Periodic Differential Equations in the Plane: A Topological Perspective 9783110551167, 9783110550405

Table of contents :
Preface
Contents
1. Periodic differential equations and isotopies
2. Massera’s theorems
3. Free embeddings of the plane
4. Index of stable fixed points and periodic solutions
5. Proof of the arc translation lemma
6. Appendix on degree theory
7. Solutions to the exercises
Bibliography
Index
De Gruyter Series in Nonlinear Analysis and Applications

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Rafael Ortega Periodic Differential Equations in the Plane

De Gruyter Series in Nonlinear Analysis and Applications

|

Editor-in Chief Jürgen Appell, Würzburg, Germany Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Mikio Kato, Nagano, Japan Wojciech Kryszewski, Torun, Poland Umberto Mosco, Worcester, Massachusetts, USA Louis Nirenberg, New York, USA Simeon Reich, Haifa, Israel Alfonso Vignoli, Rome, Italy Vicenţiu D. Rădulescu, Krakow, Poland

Volume 29

Rafael Ortega

Periodic Differential Equations in the Plane |

A Topological Perspective

Mathematics Subject Classification 2010 Primary: 37E30, 34C15, 34C25; Secondary: 37C25, 34D20 Author Prof. Dr. Rafael Ortega Universidad de Granada Facultad de Ciencias Departamento de Matemática Aplicada 18071 Granada Spain [email protected]

ISBN 978-3-11-055040-5 e-ISBN (PDF) 978-3-11-055116-7 e-ISBN (EPUB) 978-3-11-055042-9 ISSN 0941-813X Library of Congress Control Number: 2019933659 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2019 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface In the period 1945–1951, Cartwright and Littlewood published a series of papers on the forced Van der Pol circuit. Many connections between periodic differential equations and the topology of the plane were discovered in these remarkable works. In particular, some complicated dynamics was described in precise terms. Motivated by Cartwright–Littlewood’s results, in 1949 Levinson considered a modified (piece-wise linear) version of the Van der Pol circuit. Again some intricate dynamics appeared. Twenty years later, the study of Levinson’s paper inspired Smale in the construction of his famous horseshoe. Going back to the 1940s, and more precisely to 1944, we find another paper by Levinson. This time he was not concerned with a particular equation; instead he dealt with the general class of periodic differential equations in the plane. The contents included the notion of a dissipative system, discussions on the possible structure of planar attractors, and the use of degree theory to find general inequalities connecting the number of harmonic and sub-harmonic solutions. A harmonic solution is a periodic solution having the same period as the equation. Sub-harmonic solutions have larger periods. Levinson’s paper was full of new ideas but it also contained some inaccuracies that were corrected by Massera in 1949. Probably motivated by Levinson’s results, Massera asked the following question: does the existence of a sub-harmonic solution imply the existence of a harmonic solution? He found that the answer to his question is affirmative for linear periodic equations of arbitrary dimension, and also for nonlinear equations in the plane. In the process he realized that the assumption on the existence of a sub-harmonic solution could be replaced by the existence of a bounded solution. The answer to the previous question is negative for nonlinear equations in more than two dimensions. The best-known paper by Massera was published in 1950. Together with the main results, there were ingenious examples showing the sharpness of the assumptions. In the field of periodic equations, Massera’s theorems were of a new category, since they were concerned with a general class of equations; they only applied to low dimension and proofs were not obtained by traditional analytical techniques. Indeed these theorems were direct consequences of well-known facts in planar topology. The most traditional proofs in the field of differential equations use analytical tools. Proofs by topological arguments are very different and sometimes it is felt that the level of rigor is not comparable. On this matter it may be worth to recall the opinion of a master of analysis: [. . .] les raisonnements géométriques ont actuellement mauvaise presse et nombreux son ceux qui n’admettent que les raisonnements ‘arithmétisés’. Certes, tout raisonnement correct est arithmétisable, théoriquement, et cette arithmétisation fournirait une vérification; mais il n’en resulte https://doi.org/10.1515/9783110551167-201

VI | Preface

ni que l’arithmétisation soit indispensable, ni, a fortiori, qu’on doive déclarer inexact tout raisonnement qui n’est pas exprimé dans le langage analytique à la mode.1

Despite all possible distrust of topological proofs, the community of differential equation researchers accepted very soon the importance of Massera’s theorems for the line and the plane, and they were included in many textbooks, in most cases without a proof for the second theorem. This result is indeed a consequence of a lemma due to Brouwer. Let us now go back in time in order to present this old and important lemma. The dynamics of an autonomous system in the plane is not too complex and most of the features are described by Poincaré–Bendixson theory. The topological clue behind this simplicity is the Jordan curve theorem. Even if we remain in two dimensions, periodic systems may show a very intricate behavior. This fact was discovered by Poincaré and it was later confirmed in some of the papers previously mentioned. The qualitative properties of a periodic system are linked to the analysis of the iterates of a certain planar map. Therefore periodic systems lead to the kingdom of discrete dynamics. The motion of a particle can be described by a path (continuous dynamics) or by a sequence of successive positions (discrete dynamics). In a work published in 1912, Brouwer found a technique allowing one to use the Jordan curve theorem in a discrete setting. Given two consecutive positions of a particle, we connect them by a more or less arbitrary arc, then we follow the iterates of this arc. This construction, somewhat mimicking the notion of an orbit in continuous systems, led Brouwer to a fundamental discovery for discrete systems in the plane, the so-called arc translation lemma. This result says that if two iterates of the arc have a crossing, then the index of the map in an appropriate region is one. In particular, the map has at least one fixed point. This is valid for orientation-preserving embeddings, a class of maps fitting well with periodic systems. The traditional proof of the arc translation lemma consists in a direct computation of the winding number along a Jordan curve constructed by juxtaposing some iterates of the translation arc. The argument seems very convincing; a crucial property, the preservation of orientation for the map, is employed in a rather intuitive way. In 1984 Brown presented a new proof, using Alexander’s isotopy together with several arguments of deformation of maps. These deformations (isotopies) preserve the fixed point index. After several steps, there appeared a sort of canonical situation: a map having 1 A free translation: “Geometrical reasonings have nowadays a bad press. Many people only accept ‘arithmetized’ reasonings. Certainly, every correct reasoning can be arithmetized, at least in theory. This arithmetization will provide a verification. However, this does not mean that the arithmetization is essential or that, by principle, any argument not expressed in a fashionable analytic language should be considered incorrect”. Extracted from H. Lebesgue, Analyse de la thèse de M. Antoine. Bull. Sci. Math. 46 (1922).

Preface | VII

an invariant Jordan curve. In this case the computation of the index is easy and the proof is fully rigorous. Massera’s theorem is obtained as a corollary. The purpose of this book is to make accessible the theory of translation arcs to the reader interested in periodic differential equations or in discrete dynamics. Real experts in topology and dynamics may feel that the exposition is too meticulous. Also, they may argue that many deep and recent results in the area are not included. They are right in both cases. Indeed this book is addressed to a different audience, composed mainly of students and researchers coming from analysis. The approach and the choice of the results are motivated by the applications to differential and difference equations. The organization of the book is highly nonlinear, in an attempt to motivate the reader. Chapter 1 contains some initial examples, as well as general results valid for differential equations in arbitrary dimension. The links between periodic equations, isotopies and iterates of maps are discussed. Massera’s theorems and a wide collection of examples and counter-examples are presented in Chapter 2. The proof of the second theorem is postponed to the following chapter. The topological point of view is introduced in Chapter 3. It deals with the theory of planar embeddings (continuous and one-to-one maps), including the statement of Brouwer’s lemma. The last part of the chapter is devoted to applications. In particular, we prove the second Massera theorem and some fixed point theorems due to Montgomery and to Cartwright and Littlewood. The proof of Brouwer’s lemma is postponed to Chapter 5. Applications to stability theory are presented in Chapter 4. In particular we analyze the connections between index and stability of fixed points. This is a delicate question, whose answer depends upon the dimension. Brouwer’s lemma is crucial in the analysis of the two-dimensional case. Applications to populations dynamics and Hamiltonian systems are also included. Finally, in Chapter 5 we follow the elegant approach by Brown to present a complete proof of Brouwer’s lemma. The book is completed by an appendix on degree theory and the solutions of all the proposed exercises. Writing this book has been a long process. The origin goes back to a conversation with Russell A. Smith in 1988. At that time I was young and only read recent papers; it was Professor Smith who convinced me to read the original papers by Massera. Learning about Brown’s proof of the arc translation lemma was a result of my collaboration with Norman Dancer, initiated during his visit to Granada in 1992. This collaboration was a decisive step in my work. Around ten years later, teaching a course on translation arcs in Milan, I wrote some notes and Massimo Tarallo read them thoroughly. The initial idea of writing a monograph on this topic yielded an incomplete text (around 80 pages) entitled “Topology of the plane and periodic differential equations”. Rogério Martins read that version carefully and helped me with a first draft of the figures. In 2017 I received a kind letter from Dr. Damialis, proposing to publish a monograph for De Gruyter. It was time to complete the task.

VIII | Preface Pedro García Sánchez and Markus Kunze have suggested useful corrections in the final manuscript. My thanks are also to Alfonso Ruiz Herrera, who has read and corrected the successive versions of this project.

Contents Preface | V 1 1.1 1.2 1.3 1.4 1.5 1.6

Periodic differential equations and isotopies | 1 From oscillators to population dynamics | 1 The Poincaré map | 4 Diffeomorphisms and Poincaré maps | 8 Isotopies and diffeotopies | 9 The flow of a periodic equation | 11 Bibliographical remarks | 14

2 2.1 2.2 2.3 2.4 2.5

Massera’s theorems | 17 Equations in the line | 17 Some examples in the plane | 22 Second Massera theorem | 25 Persistence of harmonic solutions | 30 Bibliographical remarks | 33

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.9.1 3.9.2 3.10 3.11

Free embeddings of the plane | 37 Topological embeddings | 37 Orientation-preserving embeddings | 40 Limit sets, trivial dynamics and free embeddings | 42 Brown’s degree condition | 44 Translation arcs and Brouwer’s lemma | 46 Existence of translation arcs | 48 The complement of the fixed point set | 52 More on free embeddings and a proof of Brown’s theorem | 56 Two fixed point theorems | 59 Area-preserving maps of the open disk | 59 Fixed points on invariant continua | 63 Proof of Massera’s theorem | 68 Bibliographical remarks | 70

4 4.1 4.2 4.3 4.4 4.5

Index of stable fixed points and periodic solutions | 73 Lyapunov’s stability | 73 The index of a stable equilibrium | 78 The plug construction | 82 Some dynamical insights | 88 The index of a stable fixed point | 93

X | Contents 4.6 4.7 4.7.1 4.7.2 4.8 4.9 4.10 4.10.1 4.10.2 4.11

Simply connected invariant neighborhoods and proof of Theorem 12 | 97 Some corollaries of Theorem 12 | 101 Resonance at the roots of the unity | 101 Stability and persistence | 104 Global asymptotic stability and extinction of two populations | 106 Stable fixed points of area-preserving maps | 112 Some applications to Hamiltonian systems | 118 The number of stable periodic solutions | 119 Stable closed orbits with two degrees of freedom | 123 Bibliographical remarks | 127

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Proof of the arc translation lemma | 131 Equivalence of embeddings | 131 Compression of translation arcs | 132 Reduction to periodic orbits | 134 Lemmas on isotopies | 139 Computation of the degree on certain Jordan domains | 142 Brouwer’s lemma and some consequences | 145 Bibliographical remarks | 145

6 6.1 6.2 6.3 6.3.1 6.3.2 6.3.3 6.3.4 6.3.5 6.4 6.5 6.6

Appendix on degree theory | 147 Definition | 147 Some useful properties | 148 Computation of the degree | 150 The degree in one dimension | 150 Degree on a disk | 151 Degree of a linear map | 151 Linearization property | 152 The degree of a Cartesian product | 153 Fixed point index and some principles | 153 Degree and invariant curves | 155 Bibliographical remarks | 157

7 7.1 7.2 7.3 7.4 7.5 7.6

Solutions to the exercises | 159 Chapter 1 | 159 Chapter 2 | 161 Chapter 3 | 166 Chapter 4 | 169 Chapter 5 | 174 Chapter 6 | 175

Contents | XI

Bibliography | 179 Index | 185

1 Periodic differential equations and isotopies The motion of a fluid can be described in terms of the velocity field or by a family of maps determining the position of all particles at each instant. The first approach is associated with the theory of differential equations and the second with the topological notion of isotopy. The connections between the two concepts will be discussed. After some motivating examples in the plane we discuss general properties of periodic systems which are valid in any number of dimensions. The notation ‖ ⋅ ‖ will be reserved for the Euclidean norm, ‖x‖ = √x12 + ⋅ ⋅ ⋅ + xd2

if x = (x1 , . . . , xd ) ∈ ℝd .

1.1 From oscillators to population dynamics Periodic equations appear after forcing autonomous systems. From the second order equation ü + g(u, u)̇ = 0

(1.1)

ü + g(u, u)̇ = f (t).

(1.2)

we are led to

The function f (t) is supposed to be known and periodic, say with period T > 0. Equations (1.1) and (1.2) can be taken to be models for the motion of a particle or for the evolution of an electric circuit. In the former case u(t) is the position of the particle on a line and f (t) is a mechanical external force, in the latter case u(t) is the current and the primitive of f (t) is the electromotive force. The forced harmonic oscillator is studied in basic textbooks. It can be written as ü + cu̇ + au = f (t) with a > 0. The parameter c may be positive or zero depending on whether there is friction or not. Along the same lines, but now with nonlinear equations, we find the forced pendulum equation ü + cu̇ + a sin u = f (t)

(1.3)

and the forced Van der Pol equation ü + c(u2 − 1)u̇ + au = f (t).

(1.4)

Equation (1.3) models the motion of a particle on a vertical circle under the action of gravity and an external force which can be produced by a torque. Equation (1.4) models a circuit with nonlinear resistance. https://doi.org/10.1515/9783110551167-001

2 | 1 Periodic differential equations and isotopies Sometimes equations in the plane are obtained from systems in higher dimensions having an invariant plane. This happens, for instance, in the restricted three body problem. Assume that three bodies with masses m1 > 0, m2 > 0 and m3 = 0 are moving in the space. The primaries describe elliptic orbits on the plane x, y. The corresponding trajectories qi = qi (t), i = 1, 2, in ℝ2 × {0} are Keplerian and they are known. They are periodic functions with the same period and the center of mass is placed at the origin, so that m1 q1 + m2 q2 = 0 for each t. The motion of the third body is a solution in ℝ3 of q̈ 3 =

Gm1 Gm2 (q1 (t) − q3 ) + (q (t) − q3 ), 3 ‖q1 (t) − q3 ‖ ‖q2 (t) − q3 ‖3 2

(1.5)

where G is a positive constant. This system has six dimensions (for position and velocity) and in the special case m1 = m2 = m the z axis is invariant. In fact, when the third body is placed at this axis, the resultant of the forces acting on it also has the direction of z. See Figure 1.1. The reduced equation is z̈ +

2Gmz = 0, (z 2 + r(t)2 )3/2

(1.6)

with q3 (t) = (0, 0, z(t)) and r(t) = ‖q1 (t)‖ = ‖q2 (t)‖, the distance from the primaries to the center of mass. This periodic equation is called the Sitnikov problem.

Figure 1.1: The forces in the Sitnikov problem.

Another model where an invariant plane appears is the equation of a suspension bridge due to Lazer and McKenna. In the engineering literature, the vertical oscillations of a bridge are modeled via the beam equation utt + uxxxx = f (t, x),

x ∈ [0, L],

with boundary conditions u(t, 0) = u(t, L) = 0,

uxx (t, 0) = uxx (t, L) = 0.

1.1 From oscillators to population dynamics | 3

The vertical axis is oriented downwards and thus the positions u > 0 are below the horizontal u = 0. The function f = f1 + f2 has two components. The larger, f1 , is the weight of the bridge, while smaller one, f2 , is due to external causes like winds or waves. In a suspension bridge the cables exert a force when they are tight (u > 0) and we are led to the modified equation utt + uxxxx + au+ = f (t, x), with a > 0 and u+ = max(u, 0). See Figure 1.2.

Figure 1.2: The restoring forces of the cables.

This equation is nonlinear but the homogeneity of u+ allows for some separation of variables. Assuming f (t, x) = (1 + ϵp(t)) sin

πx , L

we can search for solutions of the type u(t, x) = v(t) sin

πx , L

and this leads to v̈ + αv+ − βv− = 1 + ϵp(t),

(1.7)

with α = a + ( πL )4 , β = ( πL )4 , v− = max(−v, 0). This is the equation of an oscillator with asymmetry (the elasticity constant is different for v > 0 and v < 0). It is entertaining to design a machine for this equation. It has two linear springs with characteristic k1 = ( πL )4 and k2 = a. The second has a stop at v = 0. See Figure 1.3. We should add some device producing the external force. Finally, we describe a third strategy to produce periodic systems. Assume that we are given an autonomous system in the plane depending on some parameters α, β, . . . The system is transformed into a periodic system by allowing a periodic dependence for the parameters α = α(t), β = β(t), . . . This is particularly meaningful in ecology,

4 | 1 Periodic differential equations and isotopies

Figure 1.3: Two springs and a stop.

for, if seasonal effects are taken into account, the parameters of a population should fluctuate in an approximately periodic fashion, the period being one year. In this way we are led to the periodic Lotka–Volterra system u̇ = u(α(t) − β(t)u − γ(t)v),

v̇ = v(δ(t) ± ϵ(t)u − ζ (t)v),

where the functions α, β, . . . , ζ are T-periodic and β, γ, ϵ, ζ are non-negative. The sign ± in front of ϵ distinguishes the prey–predator model (u is the prey) and the competition model. In the pages to follow we shall deal with a general periodic system and the emphasis will be on finding properties valid for large families of systems. The above examples will only appear in some exercises.

1.2 The Poincaré map In the rest of the chapter the number of dimensions will not play a role; therefore, we consider a system in ℝd , d ≥ 1, ẋ = X(t, x),

(1.8)

where X : ℝ × ℝd → ℝd is continuous and, for some fixed T > 0, X(t + T, x) = X(t, x),

for every (t, x) ∈ ℝ × ℝd .

In addition, it will be assumed that we have uniqueness for the initial value problem associated with (1.8). A local condition of Lipschitz type on the vector field is sufficient to guarantee uniqueness but this is far from necessary. Exercise 1. Prove that there is uniqueness for ü + u1/3 = 0 but not for ü − u1/3 = 0. Hint: use conservation of energy.

1.2 The Poincaré map |

5

Given p ∈ ℝd , the solution of (1.8) satisfying the initial condition x(0) = p will be denoted by x(t, p). This solution is defined on a maximal interval ]α, ω[ with −∞ ≤ α = α(p) < 0 < ω = ω(p) ≤ +∞. If this interval is not the whole line, then the solution must blow up in finite time. This means that ‖x(t, p)‖ → ∞ as t ↑ ω or t ↓ α if ω < ∞ or α > −∞. The general theory of differential equations says that ω = ω(p) is a lower semi-continuous function and α = α(p) is upper semi-continuous. We shall consider the set d

𝒟 = {p ∈ ℝ : x(t, p) is defined in [0, T]}.

This set can also be described as ω(p) > T and so it is open. Exercise 2. Construct an example where 𝒟 is the empty set and another example where 𝒟 is disconnected. Prove that 𝒟 is an open interval if d = 1 and that every connected component of 𝒟 is simply connected if d = 2. We define the Poincaré map P : 𝒟 ⊂ ℝd → ℝd ,

p 󳨃→ x(T, p).

See Figure 1.4.

Figure 1.4: The Poincaré map.

By uniqueness, the map P is well defined and one-to-one. The theorem on continuous dependence implies that P is continuous. A crucial property of periodic equations is that, if x(t) is a solution, then x(t + T) is also a solution. From this fact and the uniqueness we deduce that x(t + T, p) = x(t, P(p)),

(1.9)

whenever it is defined. The map P is not always onto and one observes that P(𝒟) = {p ∈ ℝd : x(t, p) is defined in [−T, 0]} = {p ∈ ℝd : α(p) < −T}. The theory of periodic differential equations can be immersed in the field of discrete dynamics via the difference equation pn+1 = P(pn ).

(1.10)

6 | 1 Periodic differential equations and isotopies

Figure 1.5: Iterations of the Poincaré map.

See Figure 1.5. Given a solution x(t, p) of (1.8), with α < m− T < 0 < m+ T < ω, as a consequence of (1.9), we see that {x(nT, p)}m− ≤n≤m+ is the solution of (1.10) with p = p0 . Most of the properties of a periodic equation can be translated to the language of maps via P. Some examples are: – all the solutions of (1.8) defined at t0 = 0 are globally defined in the future [respectively, in the past] if and only if 𝒟 = ℝd [respectively, P(𝒟) = ℝd ]; – φ(t) = x(t, p) is a T-periodic solution of (1.8) if and only if p is a fixed point of P; i. e. p ∈ 𝒟 and P(p) = p; – for each N ≥ 2, φ(t) = x(t, p) is a NT-periodic solution of (1.8) if and only if p is a periodic point of P; that is, p, P(p), . . . , P N−1 (p) ∈ 𝒟 and P N (p) = p. When the solution φ(t) admits the period NT but kT is not a period for k = 1, . . . , N − 1, then φ is called a sub-harmonic of order N. It corresponds to N-cycles of P. Given a solution x(t, p) defined in the future, the associated orbit is also defined in the future, {pn }n≥0 , p0 = p. The omega limit set of the orbit {pn } is defined as Lω (p) = {q ∈ ℝd : there exist integers nk with nk → +∞, pnk → q}, and the asymptotic behavior of a solution of the differential equation is encrypted in Lω (p). The simplest instance is a solution x(t, p) which is asymptotically T-periodic. This means that x(t, p) is well defined in the future and there exists a T-periodic solution φ(t) such that x(t, p) − φ(t) → 0,

as t → +∞.

This is equivalent to saying that {pn }n≥0 is well defined and bounded and that Lω (p) is a singleton contained in 𝒟. To prove this assertion we observe that if x(t, p) is asymptotically periodic then the sequence pn = x(nT, p) converges to φ(0) ∈ 𝒟, which is the only point in Lω (p). Conversely, assume now that {pn } is bounded and Lω (p) = {q} with q ∈ 𝒟. Then pn → q

1.2 The Poincaré map |

7

and if we define the sequence of solutions xn (t) = x(t + nT, p),

t ∈ [0, T], n = 0, 1, 2, . . . ,

we observe that xn (0) = pn converges to q. As q is in 𝒟 we apply the continuous dependence and deduce that lim x (t) n→∞ n

= φ(t) uniformly in t ∈ [0, T],

where φ(t) = x(t, q). From xn (T) = xn+1 (0) we obtain φ(0) = φ(T) and so q = φ(0) is a fixed point of P. Now it is easy to prove that φ(t) is a T-periodic solution and x(t, p) − φ(t) → 0 as t → +∞. We finish this pocket dictionary (periodic equations/maps in ℝd ) with a discussion of boundedness. A solution x(t, p) of (1.8) is bounded in the future if and only if the associated positive orbit {pn }n≥0 is well defined and contained in a compact set K with K ⊂ 𝒟. To prove this we first assume that M is a bound for the solution, that is, ‖x(t, p)‖ ≤ M

if t ≥ 0.

Define K as the closure of the set {pn : n ≥ 0}. This is a compact set and if q is a point in K then there exists σ : ℕ → ℕ such that {pσ(n) } converges to q. By the continuous dependence, x(t, pσ(n) ) → x(t, q) if t ∈ [0, ω(q)[. As x(t, pσ(n) ) = x(t + σ(n)T, p) remains in the ball of radius M, the same will happen to its limit. In other words, ‖x(t, q)‖ ≤ M

if t ∈ [0, ω(q)[.

This implies that x(t, q) cannot blow up in the future and so ω(q) = +∞. In this way we have constructed K ⊂ 𝒟. To prove the converse we start with {pn } ⊂ K ⊂ 𝒟 and use the continuity of (t, q) 󳨃→ x(t, q) on the compact set [0, T] × K. Exercise 3. Consider the linear equation ẋ = A(t)x + b(t) where A = A(t) is a d × d matrix, b = b(t) is in ℝd and both are continuous and T-periodic. Prove that the existence of a solution which is bounded in the future implies the existence of T-periodic solution. Hint: given M a d × d matrix and η ∈ ℝd , the linear system Mx = η is compatible if and only if η ⊥ Ker(M ⋆ ), where M ∗ is the transpose matrix.

8 | 1 Periodic differential equations and isotopies For equations having solutions that cannot be extended globally, one must be careful with the translation to the language of Poincaré maps. We present two curious examples for period T = 2π and dimension d = 1. The first equation is ẋ = −(sin t)x3 . It can be solved and the solutions defined at t0 = 0 are x(t, p) =

p √1 + 2p2 (1 − cos t)

,

t ∈ ℝ.

This implies that 𝒟 = ℝ and P = id. However, there are solutions which are not globally defined. An example is x(t) =

1 , √2(1 − cos t)

t ∈ ]0, 2π[.

The second equation is ẋ = x + (sin t + cos t − 1)x2 , which has the solution x(t) =

1 , 1 − sin t + e−t

t ∈ ℝ.

For p = 1/2 we observe that pn = x(2πn) converges to q = 1 and so {pn }n≥0 is a bounded sequence and Lω (p) is a singleton. Despite this fact, the solution x(t) is unbounded in [0, ∞[.

1.3 Diffeomorphisms and Poincaré maps The group of homeomorphisms of ℝd will be denoted by ℋ(ℝd ) and the identity map will be denoted by id. Given a periodic differential equation (1.8), the associated Poincaré map will be in ℋ(ℝd ) if there is uniqueness and all the solutions are defined in ]−∞, +∞[. In such a case 𝒟 = P(𝒟) = ℝd and the inverse is continuous, as shown by the formula P −1 (p) = x(−T, p),

p ∈ ℝd .

Not every map in ℋ(ℝd ) can be realized as a Poincaré map. A simple example in dimension d = 1 is the homeomorphism P = −id. To show that P is not a Poincaré map we proceed by contradiction and assume that P(p) = x(T, p) for all the solutions x(t, p) of some periodic equation. Since we are working with a scalar equation, the uniqueness implies that the flow is monotone. Then p1 > p2

is equivalent to x(t, p1 ) > x(t, p2 ) for each t ∈ ℝ.

1.4 Isotopies and diffeotopies | 9

In particular, for p > 0 we should obtain P(p) = x(T, p) > x(T, 0) = P(0). This is impossible because P(p) = −p is negative and P(0) = 0. To find the class of homeomorphisms that can be realized as Poincaré maps one needs delicate properties of the topology of the Euclidean space. We shall be less ambitious and solve a similar problem for diffeomorphisms. The group of C ∞ diffeomorphisms of ℝd will be denoted by Diff∞ (ℝd ). Diffeomorphisms appear naturally as the Poincaré maps of certain differential equations. This is the case for (1.8) if we assume that X ∈ C ∞ (ℝ × ℝd , ℝd ), and all the solutions are globally defined. In fact, the smoothness of P follows from the theorem of differentiability with respect to initial conditions and the determinant of P 󸀠 never vanishes as is shown by the identity (Jacobi–Liouville formula) T

det P 󸀠 (p) = exp{∫ divx X(t, x(t, p))dt}. 0

This formula gives additional information; not only is P a diffeomorphism but it preserves orientation. This can be understood in the sense that det P 󸀠 > 0 everywhere. The subgroup of the diffeomorphisms satisfying this additional property will be ded noted by Diff∞ ⋆ (ℝ ). d ∞ d d Theorem 1. For each P ∈ Diff∞ ⋆ (ℝ ) there exists X ∈ C (ℝ×ℝ , ℝ ) which is T-periodic in t and such that all the solutions of (1.8) are globally defined and P is the associated Poincaré map.

The vector field X is not unique. If we fix the period T and compute the Poincaré map for Xω (t, x) = (x2 , −ω2 x1 , 0, . . . , 0) then we observe that P = id as soon as ω = 2πn , n = 1, 2, . . . More generally, if we know how to construct one vector field X in the T conditions of the theorem, then it is easy to construct many others. It is enough to perform changes of variable of the type y = Φ(t, x) where Φ is T-periodic in t. To prepare for the proof of Theorem 1 we shall look at periodic equations from a topological point of view. This will be the task of the next two sections.

1.4 Isotopies and diffeotopies Let Y be a topological space and let h0 , h1 : Y → Y be two homeomorphisms. An isotopy between h0 and h1 is a continuous map H : [0, 1] × Y → Y,

(t, p) 󳨃→ H(t, p) = Ht (p)

satisfying: – H0 = h0 , H1 = h1 ; – Ht : Y → Y is a homeomorphism for each t ∈ [0, 1].

10 | 1 Periodic differential equations and isotopies The reader who is familiar with homotopy theory will recognize that an isotopy is just a homotopy with an extra condition. Using changes of scale in the interval [0, 1], it is easy to prove that the notion of isotopy defines an equivalence relation in ℋ(Y), the group of homeomorphisms of Y. As an example we consider the unit circle Y = 𝕊1 = {z ∈ ℂ : |z| = 1} and observe that the identity h0 = id and the antipode map h1 = −id are isotopic via Ht (z) = eiπt z. See Figure 1.6.

Figure 1.6: An isotopy in 𝕊1 .

Let us now assume that Y is a C ∞ manifold, in particular Y might be ℝd . Given h0 , h1 ∈ Diff∞ (Y), a diffeotopy between them is an isotopy H which is in C ∞ ([0, 1] × Y, Y) and such that Ht ∈ Diff∞ (Y) for each t ∈ [0, 1]. The notion of diffeotopy defines an equivalence relation in Diff∞ (Y) which will be denoted by h0 ≈ h1 . The proof of this fact is slightly more delicate than in the case of isotopies. The previous example of an isotopy on 𝕊1 is also a diffeotopy. As a second example we discuss how to construct diffeotopies of linear maps in ℝd . Consider the Lie group GL(ℝd ) = {L ∈ ℝd×d : det L ≠ 0}, where ℝd×d is the space of endomorphisms of ℝd or the d × d real matrices. It is well known that this group has two connected components and the component of the identity is the subgroup GL⋆ (ℝd ) = {L ∈ GL(ℝd ) : det L > 0}. As GL⋆ (ℝd ) is a smooth connected manifold, given two matrices L0 , L1 ∈ GL⋆ (ℝd ), it is possible to find a smooth path γ : [0, 1] → GL⋆ (ℝd ) joining L0 and L1 . This path defines the diffeotopy Ht (p) = γ(t)p. Next we present an extension of these ideas to nonlinear maps. d Lemma 1. Every P ∈ Diff∞ ⋆ (ℝ ) is diffeotopic to id.

1.5 The flow of a periodic equation | 11

As a corollary we obtain d ∞ d Diff∞ ⋆ (ℝ ) = {P ∈ Diff (ℝ ) : P ≈ id}.

In fact the lemma gives one of the inclusions. The other is obviously true, for if H is a diffeotopy between id and P then (t, p) 󳨃→ det Ht󸀠 (p) is a continuous function which never vanishes and therefore must remain positive. Proof. We use the equivalence ≈ and divide the proof in three steps. d Step 1: there exists P1 ∈ Diff∞ ⋆ (ℝ ) with P1 (0) = 0 and P ≈ P1 . d Given w ∈ ℝ consider the translation Tw (p) = p + w. Fix v = P(0) and consider the diffeotopy Ht = T−tv ∘ P. P1 is defined as H1 . Step 2: P1 ≈ P1󸀠 (0). Define 1 P1 (tp), t ∈ ]0, 1], Ht (p) = { t 󸀠 P1 (0)p, t = 0.

This map is smooth because P1 is smooth and P1 (0) = 0.1 To check that Ht is a diffeomorphism we observe that Ht = D−1 t ∘ P1 ∘ Dt with Dt (p) = tp if t ∈ ]0, 1]. Step 3: P1󸀠 (0) ≈ id. This can be proven using a path in GL⋆ (ℝd ) as discussed above.

1.5 The flow of a periodic equation Let us go back to equation (1.8) and assume that there is uniqueness and global existence. For each t ∈ ℝ we consider the map ϕt (p) = ϕ(t, p) := x(t, p),

p ∈ ℝd ,

where x(t, p) is the solution of (1.8). We know that ϕt belongs to ℋ(ℝd ) and ϕ0 = id, ϕT = P. This shows in particular that the Poincaré map P and the identity id are isotopic. When the field X(t, x) is smooth we can say more, namely that P and id are diffeotopic. To visualize the effect of this isotopy we go back to Figure 1.4 in Section 1.2. If we start at the plane t = 0 and then move in the direction of time until we reach t = T, then the flow deforms id into P. The periodicity of the equation is reflected on the map ϕ through the property ϕt+T = ϕt ∘ ϕT ,

for each t ∈ ℝ.

This is a consequence of (1.9). 1 Notice that at this point we would lose one derivative if P ∈ Diffk (ℝd ) with 1 ≤ k < ∞.

12 | 1 Periodic differential equations and isotopies Exercise 4. Construct a periodic equation such that ϕt and ϕT do not commute for some t ∈ ℝ. Now we have sufficient ingredients to characterize periodic flows. We go the other way and assume that ϕ : ℝ × ℝd → ℝd ,

(t, p) 󳨃→ ϕt (p) = ϕ(t, p)

is a C ∞ map satisfying: – ϕt ∈ Diff∞ (ℝd ), for each t ∈ ℝ – ϕ0 = id – ϕt+T = ϕt ∘ ϕT , for each t ∈ ℝ for some T > 0.2 We can view this map as the flow of a periodic equation. To construct the associated vector field we notice that x(t) = ϕt (p) is a solution of ̇ = x(t)

𝜕ϕ 𝜕ϕ (t, p) = (t, ϕ−1 t (x(t))) 𝜕t 𝜕t

and so we define X(t, x) =

𝜕ϕ (t, ϕ−1 t (x)). 𝜕t

It satisfies all the required conditions. We check the periodicity. From ϕ(t + T, x) = ϕ(t, ϕT (x)) we obtain 𝜕ϕ 𝜕ϕ (t + T, x) = (t, ϕT (x)) 𝜕t 𝜕t and so 𝜕ϕ 𝜕ϕ (t + T, ϕ−1 (t, ϕT ∘ ϕ−1 t+T (x)) = t+T (x)) 𝜕t 𝜕t 𝜕ϕ = (t, ϕ−1 t (x)) = X(t, x). 𝜕t

X(t + T, x) =

The notion of a continuous dynamical system appears as an abstraction of the flow of an autonomous system ẋ = X(x). Along the same line one could introduce a notion of periodic flow on a topological space Y. It would consist of a continuous map ϕ : ℝ × Y → Y,

(t, p) 󳨃→ ϕt (p)

2 In general, the map ϕ is not a continuous dynamical system. Recall that a dynamical system satisfies the stronger condition ϕt1 +t2 = ϕt1 ∘ ϕt2 , for each t1 , t2 ∈ ℝ.

1.5 The flow of a periodic equation | 13

satisfying ϕt ∈ ℋ(Y) for each t and also the second and third conditions stated above for the case Y = ℝd . In this abstract setting we observe that the identity and the map ϕT are still isotopic. We are ready to prove the result on the realization of diffeomorphisms. Proof of Theorem 1. We apply Lemma 1 and obtain a diffeotopy H between id and P, so that H0 = id and H1 = P. Next we take an increasing function χ : [0, T] → [0, 1] which is C ∞ and satisfies χ(0) = 0,

χ(T) = 1,

χ (n) (0) = χ (n) (T) = 0,

n = 1, 2, . . .

For t ∈ [0, T] and p ∈ ℝd we define ϕ(t, p) = H(χ(t), p). This is a C ∞ map such that ϕt is a diffeomorphism and ϕ0 = id, ϕT = P. Also, 𝜕tn ϕ(t, p)|t=0,T = 0,

n = 1, 2, . . .

This property will guarantee that the extension of ϕ to ℝ × ℝd is smooth. This extension is unique if the third condition for a periodic flow is satisfied. For instance, if t ∈ ]T, 2T], ϕt := ϕt−T ∘ P. The map ϕ satisfies all the conditions for a periodic flow and so there is a vector field X(t, x) realizing P as a Poincaré map. Exercise 5. Given L ∈ GL⋆ (ℝd ) there exists a linear system ẋ = A(t)x, with A a smooth T-periodic matrix, such that L is the Poincaré map. d 󸀠 Exercise 6. Assume that P ∈ Diff∞ ⋆ (ℝ ) is volume-preserving; that is, det P = 1 everywhere. Prove that the vector field X(t, x) realizing P can be chosen so that

divx X = 0. Hint: the Lie group of matrices with det L = 1 is connected. Exercise 7. The standard map in the plane is given by x1 = x + ϵy, Pϵ : { y1 = y + ϵ sin(x + ϵy), with ϵ a real parameter. Construct a periodic Hamiltonian system such that P1 is the Poincaré map. Hint: in the plane, Hamiltonian and divergence free systems are the same.

14 | 1 Periodic differential equations and isotopies

1.6 Bibliographical remarks Section 1.1: The theory of oscillators is discussed in the book by Lefschetz [72]. The forced pendulum equation was already considered by Hamel in 1922. Later this equation has been analyzed by many authors and it has become a sort of touchstone in the theory of boundary value problems and also in the theory of dynamical systems of low dimension. The survey paper by Mawhin [85] explains this evolution. It is interesting to read the original papers by Van der Pol on his circuit. For instance [128], where he solved the equation by a graphical method. The paper by Van der Mark and Van der Pol [130] is very curious; they propose an electrical model of the heart based on the nonlinear circuit. The forced equation is treated in [129], where approximate solutions are found. The mathematical analysis of the forced Van der Pol equation was initiated by Cartwright and Littlewood in a series of thorough papers; see [81] and the references therein. More recently, Levi continued this type of analysis in [74]. See also [52]. A modified version of the forced Van der Pol equation was analyzed by Levinson in [77]. This paper influenced Smale in his invention of the horseshoe, as explained in [122]. The amount of literature on the restricted three body problem is enormous. The book by Pollard [108] contains a basic introduction. The study of the Sitnikov problem was initiated by Sitnikov. He employed this equation to prove the existence of oscillatory motions in the three body problem. This was important as it implied that the seven classes in the classification of final motions given by Chazy were non-empty. This theory of final motions is discussed in [7]. In a series of three papers Alekseev [2] proved the existence of horseshoes in the Sitnikov problem. The proof of this fact can also be found in the book by Moser [90]. There are many recent papers on this equation, particularly dealing with periodic solutions. The suspension bridge model of Lazer and McKenna is presented in [69]. This model leads to attractive applications of the theory of boundary value problems for equations with “jumping” nonlinearities. This type of problems had been studied since the 1970s, starting with the work by Dancer [36] and Fučik [47]. The machine with springs was presented in [95] and it was motivated by some of the discussions in [40]. The relationship between periodic equations and the seasonal effects on a logistic population are discussed by Vance and Coddington [127]. There are many papers on periodic systems of two species in competition. The dynamics of these equations was already discussed by de Mottoni and Schiaffino [39]. The prey–predator system with periodic coefficients is more complicated dynamically. For the logistic case it resembles the dynamics of a forced oscillator with friction. In the Malthusian case it has a Hamiltonian structure and the methods of classical mechanics can be applied. See [41, 96] for more details. As regards Section 1.2, the general facts about uniqueness, existence and continuous dependence can be found in many textbooks, for instance in [56]. The books

1.6 Bibliographical remarks | 15

by Pliss [106] and by Krasnosel’skiĭ [63] are recommended to learn more about periodic systems and Poincaré maps. It is also interesting to read the book by Rouche and Mawhin [111]; the emphasis in this book is in the connections between periodic equations and functional analysis. The dictionary periodic equations/Poincaré maps can be enlarged. For instance, the relationship between quasi-periodic solutions and invariant curves is discussed in the book by Siegel and Moser [121] (see also [97]). The result stated in Exercise 3 about linear equations appears in Massera’s paper [83] as Theorem 4. It should have been known to Favard when he wrote [45]. In fact Favard’s theory can be seen as an extension of this result to linear equations with almost periodic coefficients. As regards Section 1.3, the result on realization of C ∞ diffeomorphisms as Poincaré maps (Theorem 1) can be found as Theorem 1 of Chapter V in the book by Meyer and Offin [86]. The realization of C ∞ canonical maps via Hamiltonian systems is also discussed in this book. For the class of analytic diffeomorphisms one may refer to the paper by Saulin and Treschev [116] and the references therein. The realization of orientation-preserving homeomorphisms as Poincaré maps is a very delicate question. This is related to the classical problem of approximating homeomorphisms by diffeomorphisms; see [91]. As regards Section 1.4, more information as regards isotopies can be found in the book by Moise [87]. More information on diffeotopies can be found in the book by Hirsch [57]. As regards Section 1.5, the idea of interpreting the flow of a periodic equation as an isotopy with the periodicity condition can be found in the paper by Schmitt [117]. The analytical process to construct the vector field from the flow is taken from [86]. Exercise 4 can be used to correct a misprint in page 91 of [97]. Exercises 5 and 6 show 2 that when the map P is in certain subgroups of Diff∞ ⋆ (ℝ ), then the associated vector field can be chosen with additional properties. This observation can be formulated in abstract terms via Lie groups and Lie algebras. Some comments in this direction can be found in [116]. The standard map of Exercise 7 can be seen as a symplectic discretization of the pendulum equation. This is explained in [116].

2 Massera’s theorems In this chapter we work with the periodic system ẋ = X(t, x),

x ∈ ℝd ,

(2.1)

in dimensions d = 1 and d = 2. The period T > 0 is fixed, the vector field X : ℝ × ℝd → ℝd is continuous and T-periodic in t and the uniqueness for the initial value problem will be assumed without further mention. Does the existence of a bounded solution imply the existence of a T-periodic solution? The answer is affirmative for d = 1 and d = 2, although in the latter case some additional mild assumptions are required. The discussion of this question will lead to several versions of Massera’s theorems. The proofs in two dimensions require some topological machinery and will be postponed to the next chapter.

2.1 Equations in the line This section is about equations in dimension d = 1 and the conclusion will be that in this case the dynamics is simple. To show a typical example we start with the autonomous equation ẏ = y2 − 1 and perform the change of variables x = φ(t) + y, where φ(t) is a fixed and smooth T-periodic function. The effect of this change is described in Figure 2.1. We observe that the resulting equation in x has two T-periodic solutions, x± (t) = φ(t) ± 1 and that the solutions lying below x+ are asymptotically

Figure 2.1: A periodic change of variables. https://doi.org/10.1515/9783110551167-002

18 | 2 Massera’s theorems periodic. The solutions above x+ are unbounded in the future. We shall prove that this example presents all the possible behaviors for the solutions of scalar equations. Let us recall that a solution x(t) of (2.1) is asymptotically T-periodic if it is defined in ]α, ω[ with ω = +∞ and there exists a T-periodic solution φ(t) with x(t) − φ(t) → 0

as t → +∞.

In the previous chapter we obtained a characterization of these solutions in terms of the Poincaré map. There it was assumed that x(t) was defined at time t0 = 0, but this is inessential. We can characterize asymptotically T-periodic solutions as those solutions x(t) defined in ]α, ∞[ and such that the sequence {x(nT)}n≥ν converges to some point in 𝒟. Here ν > max{0, Tα }. Next we state the so-called first Massera theorem. Theorem 2. Assume that x(t) is a solution of (2.1) which is well defined and bounded in some interval [τ, ∞[. Then x(t) is asymptotically T-periodic. Before we turn to the proof we sketch an application. Exercise 8. The evolution of a logistic population is modeled by u̇ = m(t, u)u,

u ∈ ℝ+ = [0, ∞[.

It is assumed that m ∈ C ∞ (ℝ × ℝ+ ) is T-periodic in t and m(t, u) ≤ 0

if u ≥ 3.

Define extinction and prove the alternative: extinction or existence of a positive T-periodic solution. The property of scalar equations that is behind the previous theorem is the monotonicity of the flow. This means that if x1 (t) and x2 (t) are solutions defined in a common interval I and x1 (t0 ) < x2 (t0 ) for some t0 ∈ I,

then x1 (t) < x2 (t) for every t ∈ I.

This follows from uniqueness. We present a consequence for periodic equations. Lemma 2. Assume that x(t) is a solution of (2.1) defined in [τ, ∞[. Then the sequence {x(nT)}n∈ℤ,n≥ν , ν = Tτ , satisfies one of the alternatives: (i) {x(nT)} is strictly monotone; (ii) {x(nT)} is constant. Moreover, in case (ii) the solution is T-periodic.

2.1 Equations in the line

| 19

̃ = x(t+T), t ∈ [τ−T, ∞[. If x(τ) = x(τ+T) then x(t) and Proof. Consider the solution x(t) ̃ satisfy the same initial condition at time τ and so they must coincide. This implies x(t) that x(t) can be extended as a T-periodic solution and we have case (ii). Assume now that x(τ) and x(τ + T) are different, say x(τ) < x(τ + T). The monotonicity of the flow ̃ > x(t) whenever they are defined. In particular x((n + 1)T) = x(nT) ̃ implies that x(t) > x(nT) and the sequence is increasing. We are ready to present a proof of Massera’s theorem. Proof of Theorem 2. We will work with a solution that is well defined and bounded in [0, ∞[. This is not a relevant restriction for otherwise we would change the independent variable, s = t − τ. Once we know that x(t) is bounded in [0, ∞[ we translate this property to the language of Poincaré maps and observe that the sequence {x(nT)} = {P n (p)}, p = x(0), is contained in some compact set K ⊂ 𝒟. As this sequence is monotone it will converge to some point in K, and hence in 𝒟. This proves that the limit set Lω (p) is a singleton in 𝒟, and we know that this is equivalent to saying that x(t) is asymptotically T-periodic. The previous proof can be presented in the language of difference equations. Assume that 𝒟 denotes an open interval of ℝ and P : 𝒟 → ℝ is a continuous and increasing function. The dynamics of the difference equation xn+1 = P(xn ) can be understood graphically. See Figure 2.2.

Figure 2.2: Convergence to fixed points.

Assume now that {xn }n≥0 is a well defined solution and there exists a compact set K ⊂ 𝒟 such that xn ∈ K for each n ≥ 0. Then it is easy to prove that P has a fixed point q with xn → q. Exercise 9. Prove this statement and deduce Theorem 2 from it.

20 | 2 Massera’s theorems To finish the discussions on Theorem 2 we will show that it can be obtained as a corollary of the Poincaré–Bendixson theorem. This point of view is less elementary and requires some knowledge of the geometric theory of differential equations. However, it will be worth invoking it as it helps to gain some insight on the connections between periodic and autonomous equations. We start with the quotient group 𝕋 = ℝ/2πℤ, whose elements are called angles and denoted by θ = θ + 2πℤ, θ ∈ ℝ. The space S = 𝕋×ℝ is an abstract surface with coordinates (θ, x). As it is diffeomorphic to a cylinder we can visualize it as an open annulus with inner and outer boundaries corresponding to x = −∞ and x = +∞. Equation (2.1) will be interpreted as the autonomous system in S, 2π θ̇ = , T

ẋ = X(θ, x).

(2.2)

This system has no equilibria and the segment θ = constant is a global section. This means that it intersects transversally all orbits. We claim that every closed orbit meets the section exactly at one point. Otherwise we could find two consecutive points of intersection, say p0 and p1 , and consider the Jordan curve obtained by juxtaposing the arc joining p0 and p1 through the orbit with the segment joining these points through the section θ = constant. This Jordan curve separates the annulus in two regions, one of them is positively invariant under the flow while the other is negatively invariant. See Figure 2.3. After passing through p1 the orbit enters the positively invariant region. This implies that it is not possible to return to p0 ; therefore the orbit cannot be closed.

Figure 2.3: The shaded region is negatively invariant.

This property of unique intersection1 implies that the closed orbits of (2.2) correspond to the T-periodic solutions of (2.1). In particular equation (2.1) has no sub-harmonic solutions. We are ready to sketch a new proof of Theorem 2. Assume that γ + is an orbit of (2.2) which eventually enters and remains in some compact subset of the annulus. The Poincaré–Bendixson theorem says that there is a limit cycle attracting this orbit. See Figure 2.4. These conclusions are easily translated to the language of periodic 1 Not valid in higher dimensions.

2.1 Equations in the line

| 21

Figure 2.4: A geometric view of Massera’s first theorem.

equations and they become Massera’s theorem. In the next exercise we show that the uniqueness is essential for the validity of Theorem 2. Exercise 10. Construct a solution of ẋ =

π (cos t)√|1 − x2 | 2

which lies in [−1, 1], but which is not asymptotically 2π-periodic. Hint: x(t) will take the values x = 1 and x = −1 on intervals of the type [k1 π, k2 π]. The statement of Theorem 2 is not the exact version originally written by Massera. He just stated that the existence of a bounded solution implies the existence of a T-periodic solution. This weaker version does not require uniqueness. Exercise 11. Assume that X ∈ C(ℝ2 , ℝ2 ) is T-periodic in t and ẋ = X(t, x) has a solution that is bounded in [τ, ∞[. Prove the existence of a T-periodic solution without assuming uniqueness of the initial value problem. Hint: prove first the existence of a solution x(t) bounded in ]−∞, +∞[. It is obtained as a limit of translates. Next consider the solution φN (t) = sup{x(t +nT) : n = 0, ±1, . . . , ±N} and prove that limN→∞ φN is periodic. The monotonicity of the flow is also useful in the study of unbounded solutions. We present a result in this direction. Proposition 1. Assume that all the solutions of (2.1) can be extended to the whole real line. Then every solution that is unbounded in the future will also satisfy lim |x(t)| = ∞.

t→+∞

Proof. Let x(t, t0 , x0 ) denote the solution of (2.1) with x(t0 ) = x0 . The map (t, t0 , x0 ) ∈ ℝ3 󳨃→ x(t, t0 , x0 ) ∈ ℝ is continuous and so [0, T] × [0, T] × [−μ, μ] is mapped onto some compact interval [−M, M] with M = M(μ). This implies that, for any solution y(t), the following property holds: If min |y(t)| ≤ μ, [0,T]

then max |y(t)| ≤ M. [0,T]

(2.3)

22 | 2 Massera’s theorems We prove the proposition by a contradiction argument. Assume that |x(t)| does not converge to infinity and so there exist μ > 0 and tn → +∞ with |x(tn )| ≤ μ. We rewrite tn as tn = τn + σn T with τn ∈ [0, T], σn ∈ ℤ, σn → +∞. The estimate (2.3) implies that |x(σn T)| ≤ M and so the sequence {x(nT)} has a bounded subsequence. We can invoke Lemma 2 and observe that x(nT) is monotone. Thus x(nT) is bounded and the same should happen to the solution x(t) in the future. The condition of global existence is needed. The second example at the end of Section 1.2, in Chapter 1, dealt with a solution satisfying 1 lim inf x(t) = , t→+∞ 2

lim sup x(t) = +∞. t→+∞

Exercise 12. Assume that x(t) is a maximal solution of (2.1) defined in ]α, ω[. Prove that lim inf x(t) = −∞ t↑ω

and

lim sup x(t) = +∞ t↑ω

cannot occur simultaneously. Hint: if ω = ∞ and x(0) < x(T) then inf[0,∞[ x(t) = min[0,T] x(t).

2.2 Some examples in the plane We shall show that the results of the previous section do not extend to dimension d = 2. We start with the forced linear oscillator ÿ + ω2 y = sin t,

ω > 0,

which is transformed into a system of the type (2.1) with d = 2, x1 = y, x2 = y.̇ For many values of the frequency ω, this system has bounded solutions that are not asymptotically periodic with period T = 2π. This shows that Theorem 2 is not valid for d = 2. Indeed, the motions are bounded if ω ≠ 1 and they are given by y(t) =

ω2

1 sin t + c1 cos ωt + c2 sin ωt, −1

c1 , c2 ∈ ℝ.

When ω is an integer, all the solutions are T-periodic but if ω ∈ ℚ \ ℤ, say ω = pq in reduced form, then the solutions with |c1 | + |c2 | > 0 are sub-harmonic of order q. When ω ∈ ̸ ℚ these solutions are not asymptotically periodic of any period. This assertion can be justified with some Fourier analysis. Given a continuous function f : [0, ∞[ → ℝ and λ ∈ ℝ, we define T

1 f ̂(λ) = lim ∫ e−iλt f (t)dt, T→+∞ T 0

whenever this limit exists.

2.2 Some examples in the plane

| 23

Exercise 13. Assume that f is continuous and asymptotically periodic for some period p > 0. Then f ̂(λ) exists for each λ ∈ ℝ and f ̂(λ) = 0 if λ ∈ ̸ ωℤ with ω = 2π . p For the solutions of the linear oscillator with ω ∈ ̸ ℚ, we obtain ̂ = y(1)

1 , 2i(ω2 − 1)

1 ̂ y(ω) = (c1 − c2 i). 2

The numbers 1 and ω cannot coexist in the same cyclic group pℤ and so we conclude that y(t) is not asymptotically periodic. The solution with c1 = c2 = 0 is always T-periodic and so we could still conjecture that the existence of a bounded solution should imply the existence of a T-periodic solution. The rest of this section will be devoted to the construction of an example showing that even this weaker version of Massera’s principle is false in two dimensions. We shall construct a nonlinear system2 with period T = π having bounded solutions but no π-periodic solutions. The leading idea will be to find a system with Poincaré map P = −id and domain of definition 𝒟 = 𝒟+ ∪ 𝒟− with 𝒟+ ⊂ ]0, ∞[ × ℝ,

𝒟− ⊂ ]−∞, 0[ × ℝ,

P(𝒟) ∩ 𝒟 ≠ 0.

See Figure 2.5.

Figure 2.5: Dynamics of an involution.

Let us discuss the behavior of the solutions x(t, p) defined at t0 = 0. The solutions with initial condition in the shaded region ℝ2 \ 𝒟 will blow up in finite time. The same will happen when p ∈ 𝒟 but −p ∈ ̸ 𝒟. Finally, if p and −p belong to 𝒟, then u(t, p) will be periodic with period 2π (a sub-harmonic of the second order). To achieve this we start with an autonomous system in the plane u̇ = F(u),

u ∈ ℝ2 ,

2 In view of Exercise 3 of Chapter 1, the nonlinearity is essential.

(2.4)

24 | 2 Massera’s theorems such that E+ = (1, 0) and E− = (−1, 0) are equilibria, the vertical axis γ0 = {0} × ℝ is an orbit and all the solutions corresponding to this orbit blow up before time π. This means that the length of the maximal interval satisfies ω − α ≤ π. It is also assumed that F is smooth and has the symmetry F(u) = F(−u)

for each u ∈ ℝ2 .

An example is the Hamiltonian system u̇ 1 = −

𝜕H , 𝜕u2

u̇ 2 =

𝜕H 𝜕u1

with H(u1 , u2 ) = u1 (1 − 31 u21 )(1 + u22 ). Let u(t, u0 ) denote the solution of (2.4) with u(0) = u0 . We define ΩF = {u0 ∈ ℝ2 : u(t, u0 ) is defined in [0, π]} and observe that γ0 ∩ ΩF = 0,

E+ , E− ∈ ΩF .

Next we consider the periodic system (with period 2π) v̇ = π(cos t)F(v),

v ∈ ℝ2 .

(2.5)

This system is not π-periodic but the vector field G(t, v) = π(cos t)F(v) satisfies G(t + π, v) = −G(t, v) = −G(t, −v).

(2.6)

The solutions of (2.5) defined at t0 = 0 are given by v(t, v0 ) = u(π sin t, v0 ), whenever this formula is well defined. The set ΩG = {v0 ∈ ℝ2 : v(t, v0 ) is defined in [0, π]} coincides with ΩF . The searched system is obtained from (2.5) via the change of variables x = R[t]v,

R[t] := (

cos t − sin t

sin t ). cos t

̇ = JR[t] with J = ( 0 1 ), we obtain As R[t] −1 0 ẋ = X(t, x) := Jx + R[t]G(t, R[−t]x).

(2.7)

2.3 Second Massera theorem

| 25

From (2.6) and R[t + π] = −R[t], it follows that this system has period T = π. By construction x(t, p) = R[t]v(t, p) = R[t]u(π sin t, p), and so 𝒟 = ΩG = ΩF ,

P(p) = x(π, p) = −p.

The reader may have the impression that the previous construction is rather artificial. However, the resulting system can be chosen as a Hamiltonian system that is polynomial in x and trigonometric in t. Namely, 1 h(t, x1 , x2 ) = − (x12 + x22 ) + π(cos t)H(R[−t]x), 2 where H is the polynomial function defined to produce an example of system of the type (2.4). It would be interesting to know if a similar example can be obtained for a Newtonian equation. Exercise 14. For fixed T > 0 and d = 1, 2, . . ., consider the Banach spaces d

𝒜 = {x ∈ C(ℝ, ℝ ) : x(t + T) = −x(t), d

𝒫 = {x ∈ C(ℝ, ℝ ) : x(t + T) = x(t),

t ∈ ℝ}, t ∈ ℝ},

endowed with the uniform norm ‖ ⋅ ‖∞ . For d = 2 construct a linear isomorphism Φ : 𝒜 → 𝒫 satisfying ‖Φx‖∞ = ‖x‖∞ ,

‖Φx‖1 = ‖x‖1 ,

x ∈ 𝒜.

Prove that a similar isomorphism does not exist if d = 1. T The definition of the norm is ‖x‖∞ = maxt∈[0,T] ‖x(t)‖, ‖x‖1 = ∫0 ‖x(t)‖dt. Hint: consider Φ−1 (x) with x ≡ constant.

2.3 Second Massera theorem In this section we discuss the properties of the system (2.1) when d = 2. The basic assumption will be the global existence for the future of the solutions defined at t0 = 0, that is, 2

𝒟=ℝ .

(2.8)

We observe that this assumption excludes the second example of the previous section. The proofs are postponed to Section 3.10 in the next chapter, after some topological tools have been developed. Let us start with the classical version of Massera’s result.

26 | 2 Massera’s theorems Theorem 3. Assume that (2.8) holds and there exists a solution of the system (2.1) that is bounded in the future; then there exists a T-periodic solution. The example in the previous section shows that (2.8) is essential. Also, the dimension plays a role. This is illustrated with the following example in dimension d = 3, where all the solutions are defined in ]−∞, +∞[, some are bounded and none of them is periodic. Fix T = 2π and consider the system ÿ + ω2 y = 0,

ż = sin t + (1 − ẏ 2 − ω2 y2 ),

where ω > 0. This system can be expressed in the form (2.1) with x1 = y, x2 = ẏ and x3 = z. The quantity 2h = ẏ 2 + ω2 y2

(2.9)

is a first integral and all the solutions are globally defined. In the space x1 , x2 , x3 equation (2.9) defines a cylinder with elliptical basis for each h > 0. When h = 0 the cyliṅ z(t)) in Figure 2.6. The only der degenerates to a line. We sketch the curves (y(t), y(t), bounded solutions (in the future or in the past) are those at the cylinder 2h = 1. The explicit formula for these solutions is y(t) = c1 cos ωt + c2 sin ωt,

z(t) = − cos t + k,

with ω2 (c12 + c22 ) = 1 and k ∈ ℝ. If ω ∈ ̸ ℚ these solutions are not periodic for any period. Use Exercise 13.

Figure 2.6: Three possible dynamics.

Exercise 15. Assume that d ≥ 2 and the system (2.1) has a bounded solution x(t) = (x1 (t), . . . , xd (t)) such that the coordinates x1 (t), . . . , xd−1 (t) are T-periodic. Prove the existence of a T-periodic solution. We go back to dimension d = 2. The proof of Theorem 3 will give additional information. Indeed, the bounded solution can be replaced by a solution satisfying lim inf ‖x(t)‖ < ∞, t→+∞

2.3 Second Massera theorem

| 27

when all the solutions are well defined in the future. This leads to the following variant of Massera’s result. Theorem 4. Assume that every solution of (2.1) is defined in the future and there are no T-periodic solutions, then lim ‖x(t, p)‖ = ∞ for every p ∈ ℝ2 .

t→+∞

It must be noticed that the condition on global existence in this result is stronger than (2.8). Now it is assumed that the maximal interval of every solution of (2.1) is of the type ]α, +∞[, with −∞ ≤ α < +∞, and this may include solutions which are not defined at time t0 = 0. We illustrate this subtle difference with an example. See also the example at the end of Section 1.2. Consider the scalar equation ẋ = f (t, x)

(2.10)

with f (t, x) = {

(cos t − sin t − 1)x2 + x, 0,

x ≥ 0, x < 0.

The solution satisfying x(0) = p is p

x(t, p) = { p(1−cos t)+e p,

−t

,

p ≥ 0,

p < 0,

(2.11)

and it is defined in ]−∞, +∞[. In particular, 𝒟 = ℝ. On the other hand, the function 1 x(t) = 1−cos is another solution defined on the maximal interval ]0, 2π[ and therefore t the condition on global existence in Theorem 4 does not hold. The previous example can be modified to show that the condition (2.8) cannot replace the condition on global existence in Theorem 4. Before doing this we make some additional observations on equation (2.10). Let us fix a number δ ∈ ]0, 41 [, then f (t, δ) > 0 for each t ∈ ℝ. The graph in Figure 2.7 roughly describes the behavior of the solution x(t, p) with p ≥ δ. Looking at this graph or just by direct computation we observe that the solution of (2.10) satisfying x(t0 ) = δ with t0 ≥ 0 is well defined on [t0 , +∞[. We are ready to modify the equation. Define f (t, x), fδ (t, x) = { f (t, δ),

x ≥ δ, x < δ,

and consider the system ẋ1 = fδ (t, x1 ),

ẋ2 = 0.

(2.12)

28 | 2 Massera’s theorems

Figure 2.7: A schematic view of the solutions.

We claim that the condition (2.8) holds. Indeed, given any solution (x1 (t), x2 (t)) defined at time t0 = 0, x1 (t) is increasing as long as it remains in {x1 ≤ δ}. Outside this region it must coincide with a solution of (2.10). This proves that 𝒟 = ℝ2 and, from the same properties, it is easy to show that the system (2.12) has no periodic solutions. However, the solution (x1 (t), x2 (t)) = (x(t, δ), 0) satisfies 󵄩 1 󵄩 lim inf 󵄩󵄩󵄩(x1 (t), x2 (t))󵄩󵄩󵄩 = , t→+∞ 2

󵄩 󵄩 lim sup 󵄩󵄩󵄩(x1 (t), x2 (t))󵄩󵄩󵄩 = +∞. t→+∞

Exercise 16. Construct a system in the conditions of Theorem 4 and such that the convergence limt→+∞ ‖x(t, p)‖ = ∞ is not uniform in ‖p‖ ≤ 1. The previous results can be adapted to systems which are not defined in the whole plane. Assume that Ω is an open and connected subset of ℝ2 and consider the system ẋ = X(t, x),

x ∈ Ω,

where X : ℝ×Ω → ℝ2 is continuous and T-periodic in t and there is uniqueness for the initial value problem. If we also assume that Ω is simply connected, then it is possible to find a C ∞ diffeomorphism3 Φ from ℝ2 onto Ω. The change of variables x = Φ(y) transforms the system into ẏ = Φ󸀠 (y)−1 X(t, Φ(y)),

y ∈ ℝ2 .

We can apply the previous results to the system in y and then transport them to the system in x. The assumption (2.8) on global existence becomes: – for each p ∈ Ω the solution x(t, p) is well defined and remains in Ω for each t ≥ 0. The existence of a bounded solution now reads: – there exist a compact set K ⊂ Ω and a solution x(t) defined in [0, ∞[ and such that x(t) ∈ K for each t ≥ 0. 3 Riemann’s theorem on conformal mappings can be applied.

2.3 Second Massera theorem

| 29

Under these conditions we can apply Theorem 3 and deduce the existence of a T-periodic solution lying in Ω. An important example originates with the search of positive periodic solutions for Kolmogorov systems. These are systems of the type ẋi = xi Fi (t, x1 , x2 ),

xi > 0, i = 1, 2,

(2.13)

with Fi T-periodic in t and continuous. It is also assumed that there is uniqueness for (2.13). The domain Ω = ]0, ∞[ × ]0, ∞[ is diffeomorphic to ℝ2 via x1 = ey1 , x2 = ey2 . Moreover, Ω is invariant for (2.13) and we can say that (2.13) has a positive T-periodic solution if all the solutions are globally defined in the future and it has a solution satisfying 0 < m ≤ x(t) ≤ M < ∞,

i = 1, 2,

for each t ∈ [0, ∞[. Exercise 17. Consider the prey–predator system {

u̇ 1 = u1 (a(t) − b(t)u1 − c(t)u2 ), u̇ 2 = u2 (d(t) + e(t)u1 − f (t)u2 ),

u1 ≥ 0,

u2 ≥ 0,

where the functions a, b, . . . , e, f are continuous and 1-periodic and b, c, e, f are positive. Prove the exclusion principle: in the absence of coexistence states (positive 1-periodic solutions) there is pseudo-extinction. This means that every solution satisfies min{u1 (t), u2 (t)} → 0

as t → +∞.

In the above discussions the domain Ω was simply connected and this is essential. Massera’s principle is not valid for arbitrary open subsets of the plane or for general surfaces. We illustrate this in the case of the cylinder, which is diffeomorphic to an open annulus of the plane. Consider the manifold S = 𝕋 × ℝ,

𝕋 = ℝ/2πℤ

with coordinates (θ, z), θ = θ + 2πℤ, θ ∈ ℝ. A periodic equation in S can be represented by its lift to the plane θ̇ = F(t, θ, z),

ż = G(t, θ, z),

where the functions F and G are T-periodic in t and 2π-periodic in θ. The family of solutions in ℝ2 , (θ(t) + 2πn, z(t)), n ∈ ℤ, corresponds to one solution in the cylinder which is denoted by (θ(t), z(t)). With this interpretation, a solution in the cylinder is T-periodic if θ(t + T) = θ(t) + 2πn,

z(t + T) = z(t),

t ∈ ℝ,

30 | 2 Massera’s theorems for some n ∈ ℤ. A solution (θ(t), z(t)) is bounded if the coordinate z(t) is bounded. Consider the system θ̇ = ω,

ż = sin t,

T = 2π,

with solutions θ(t) = θ0 + ωt,

z(t) = z0 − cos t.

All the solutions are bounded in the cylinder and they are periodic of period 2πq if ω = pq . When ω is irrational there is no periodic solution. Probably the reader has noticed the connection with the example in 3d which was presented at the beginning of the section. Exercise 18. Consider the forced pendulum equation ẍ + cẋ + sin x = β sin t,

β ∈ ℝ, c > 0,

and interpret it as an equation on the cylinder, θ = x, z = x.̇ Assume that there exists a solution satisfying 󵄨󵄨 p 󵄨󵄨󵄨 󵄨 sup 󵄨󵄨󵄨x(t) − t 󵄨󵄨󵄨 < ∞ q 󵄨󵄨 t≥0 󵄨󵄨 and prove the existence of a periodic solution in the cylinder. Compute its period. Hint: Θ = θ − pq t.

2.4 Persistence of harmonic solutions The first question posed by Massera in his original work was: does the existence of periodic solutions (of any period) imply the existence of at least one harmonic vibration? By a harmonic vibration he understood a T-periodic solution and we know from the previous section that the answer is affirmative if 𝒟 = ℝ2 . In this section we improve this conclusion in the following sense: if there is a periodic solution which is not of period T, then small perturbations of the system (2.1) will also have a harmonic vibration. To make this statement precise we introduce the following. Definition 1. The system (2.1) has persistence of T-periodic solutions if there exist a number δ > 0 and a compact set K ⊂ ℝ2 such that if X ⋆ : ℝ × ℝ2 → ℝ2 is continuous, T-periodic in t and ‖X(t, x) − X ⋆ (t, x)‖ ≤ δ

for each (t, x) ∈ ℝ × K,

then the system ẋ = X ⋆ (t, x) has at least one T-periodic solution lying in K.

(2.14)

2.4 Persistence of harmonic solutions | 31

We abbreviate the terminology by saying that the system is T-persistent. A first observation about this notion is that it can be assumed that there is uniqueness for the initial value problem associated to (2.14). If this were not the case, we could approximate X ⋆ by smooth vector fields and then obtain a periodic solution via Ascoli’s theorem. The basic tool to prove persistence is degree theory. Now we present two examples where persistence can be proved simply using Brouwer’s fixed point theorem. The first example is the linear system ẋ = −x,

x ∈ ℝ2 ,

with any prescribed period T > 0. We prove the persistence with δ = 1 and K = {x ∈ ℝ2 : ‖x‖ ≤ 1}. To check this we consider the perturbed system ẋ = −x + R(t, x) and assume that R is such that there is uniqueness and ‖R(t, x)‖ ≤ 1

if ‖x‖ ≤ 1.

The vector field X ⋆ (t, x) = −x + R(t, x) points inwards at the boundary of K, that is, ⟨X ⋆ (t, x), n(x)⟩ ≤ 0

if ‖x‖ = 1,

where n(x) = x/‖x‖. This implies that K is positively invariant and so the solutions satisfy ‖x(t, p)‖ ≤ 1

if t ≥ 0 and ‖p‖ ≤ 1.

In particular the Poincaré map is well defined on K and P(K) ⊂ K. The fixed point of P obtained via Brouwer’s theorem corresponds to a solution lying in K. A more subtle example of persistence is provided by the autonomous system ẋ = X(x) := [

2

1 − ‖x‖2 ] (E − x) 1 + ‖x‖2

with E = (1, 0). All the points in the unit circle are equilibria and the phase portrait is drawn in Figure 2.8. We fix a period T > 0 and observe that the set of T-periodic solutions is just the set of equilibria. To prove the persistence we work with K = {x ∈ ℝ2 : ‖x − E‖ ≤ 3} and observe that 2

[

1 − ‖x‖2 9 ] ≥ 289 1 + ‖x‖2

if ‖x − E‖ = 3.

32 | 2 Massera’s theorems

Figure 2.8: A continuum of equilibria.

Given a perturbed field, ⟨X(x) + R(t, x), n(x)⟩ ≤ −

81 + ‖R(t, x)‖ 289

x−E if ‖x − E‖ = 3 and n(x) = ‖x−E‖ . This shows that K is positively invariant if ‖R‖ is small and so there is persistence. It is easy to present examples of systems which are not T-persistent and have periodic solutions. The equation

ẍ + sin x = 1 has the equilibria x ≡ π2 +2nπ, n = 0, ±1, ±2, . . . , which can be interpreted as T-periodic solutions. The perturbed equation ẍ + sin x = 1 + δ(t) has no T-periodic solutions if δ(t) > 0 everywhere. Indeed, for any solution, T

T

̇ ̇ ̈ x(T) − x(0) = ∫ x(t)dt = ∫(1 + δ(t) − sin x(t))dt > 0. 0

0

Isochronous systems are another class of non-persistent systems. A system (2.1) will be called T-isochronous if all the solutions are T-periodic. In particular they satisfy 𝒟 = ℝ2 and P = id. We prove that these systems are not persistent using the ideas of Section 1.5 in Chapter 1. To this end we assume that X is C ∞ and the associated periodic flow {ϕt }t∈ℝ satisfies X(t, x) =

𝜕ϕ (t, ϕ−1 t (x)). 𝜕t

Consider the family in Diff∞ (ℝ2 ), ψt (p) = ϕt (p + tv)

2.5 Bibliographical remarks | 33

where v ∈ ℝ2 , v ≠ 0. It satisfies ψ0 = id and ψt+T = ψt ∘ ψT . To obtain the second identity we have used the idea that ϕ is a periodic flow and ϕT = id. Now we know that {ψt } is a periodic flow associated to the periodic equation ẋ = X ⋆ (t, x) :=

𝜕ψ (t, ψ−1 t (x)). 𝜕t

This equation has no periodic solutions because the Poincaré map is a translation, namely ψT (p) = ϕT (p + Tv) = p + Tv. We shall prove that ‖X(t, x) − X ⋆ (t, x)‖ → 0 as v → 0, uniformly on compact sets. From the definition of ψ, −1 ψ−1 t (p) = ϕt (p) − tv,

𝜕ϕ 𝜕ϕ 𝜕ψ (t, p) = (t, p + tv) + (t, p + tv)v. 𝜕t 𝜕t 𝜕p

This implies that X(t, x) − X ⋆ (t, x) = −

𝜕ϕ (t, ϕ−1 t (x))v 𝜕p

and the conclusion follows. Exercise 19. Construct an example of a system which is not T-isochronous but 𝒟 = ℝ2 and P = id. After these discussions on the notion of persistence we state the result that we had in mind at the beginning of the section. Theorem 5. Assume that 𝒟 = ℝ2 and the system (2.1) has a periodic solution that is not of period T. Then the system is T-persistent. Again the proof is postponed to Chapter 3. Exercise 20. Given σ > 0, construct a T-periodic system (2.1) having a periodic solution with minimal period σ. Exercise 21. Prove that if (2.1) has a σ-periodic solution φ(t) and σ ∈ ̸ Tℚ, then the function τ ∈ ℝ 󳨃→ X(τ, φ(t0 )) is constant for each t0 ∈ ℝ.

2.5 Bibliographical remarks The contents of Section 2.1 can be seen as dealing with variations of the first Massera theorem, also called the convergence Massera theorem (CMT). Results along this line were first presented in [83], published in 1950. Later they were included in two books

34 | 2 Massera’s theorems [106, 136]. Theorem 2 also appeared in a paper by Amerio [5], published also in 1950. Amerio’s paper was concerned with an autonomous second order equation (a forced pendulum with constant torque) and the result for first order periodic equations was just an auxiliary lemma. The deep connection between CMT and Poincaré–Bendixson theory had already been mentioned in [83]. The more elementary proofs based on the monotonicity of the flow are in the origin of the theory of monotone dynamical systems [37, 58]. This theory has important applications in population dynamics because cooperative and competitive species lead to monotone flows, either in the future or in the past. There are some extensions of CMT to periodic systems of higher dimension (d ≥ 2). In [123] R. A. Smith introduced a generalized quadratic Lyapunov function and found a class of systems having the property of convergence to periodic solutions. The idea underlying the proof is the construction of a curve which is invariant under the Poincaré map such that all the orbits of P are attracted by it. After this construction, the strategy of the proof of Theorem 2 is adapted to this curve. This idea of reduction to onedimensional dynamics also appears in the study of monotone systems. The connection between the two theories is explained in [115]. There is also a large amount of literature adapting CMT to other settings (first order partial differential equations, delay and stochastic equations, non-autonomous equations with general recurrence properties, . . .). The very clever example in Section 2.2 is taken from [83]. It seems that the construction of this example played an important role in Massera’s research. Motivated by work by Levinson [76], he studied in [82] the number of sub-harmonic solutions for periodic dissipative systems in the plane. Given such a system, N(q) denotes the number of periodic solutions of period qT not having lower periods of the type kT with 1 ≤ k < q. Some properties of the family of sequences {N(q)}q≥1 are analyzed in [82]. For dissipative systems it is well known that N(1) > 0, but in the class of general systems the case N(1) = 0 cannot be excluded. This suggests the following question for the class of general periodic systems in the plane: does the inequality N(q) > 0 for some q ≥ 2 imply N(1) > 0? I conjecture that this was the initial motivation leading to [83]. The example shows that the answer is negative, at least if some solutions can blow up in finite time. As regards Section 2.3, Theorem 3 is usually considered the main result of [83]. The stronger theorem, Theorem 4, is not mentioned by Massera, although he was probably aware of it. It is an useful refinement because it describes the dynamics in the absence of harmonic solutions: all orbits go to infinity. Again, there are extensions of Theorem 3 to certain systems in higher dimensions. In [119] Sell considered a class of autonomous systems in four dimensions having a skew-product structure. He showed that the existence of a bounded orbit implies the existence of a closed orbit. In [123], already mentioned, R. A. Smith imposed a second condition to his Lyapunov function in order to reduce the dynamics to a surface (the amenable manifold). Then he adapted the proof of Theorem 3 to this surface. In [28]

2.5 Bibliographical remarks | 35

Campos obtained a new theorem of Massera type for monotone periodic systems in three dimensions. This time the reduction of the number of dimensions was obtained with the ideas of monotone flows. Massera’s theorems have many implications in the class of Kolmogorov systems, defined on the first quadrant. In this context the paper by Zanolin [137] is very informative. Section 2.4 takes some ideas from the paper [26]. Exercise 21 can be found as a theorem in [106]. See also the recent paper [34].

3 Free embeddings of the plane Given a vector field in the plane, X : ℝ2 → ℝ2 , and a closed orbit Γ of the associated system ẋ = X(x), it is well known that there exists an equilibrium lying inside Γ, X(x∗ ) = 0. The parallel result in discrete dynamics is Brouwer’s lemma. Instead of a vector field and a differential equation we have a map h : ℝ2 → ℝ2 and a difference equation xn+1 = h(xn ). The closed orbit is replaced by a finite union of iterates of an arc α, α ∪ h(α) ∪ ⋅ ⋅ ⋅ ∪ hn (α), separating the plane. Now the conclusion is the existence of a fixed point x∗ = h(x∗ ). See Figure 3.1.

Figure 3.1: Continuous vs. discrete.

Notice that the analogy between Γ and α ∪ h(α) ∪ ⋅ ⋅ ⋅ ∪ hn (α) is not complete. In most cases the set α ∪ h(α) ∪ ⋅ ⋅ ⋅ ∪ hn (α) will not be invariant under h. The precise statement of Brouwer’s lemma and some of its many dynamical applications are presented in this chapter. The proof is postponed to Chapter 5.

3.1 Topological embeddings A map h : ℝ2 → ℝ2 which is continuous and one-to-one will be called a topological embedding. The class of topological embeddings will be denoted by ℰ (ℝ2 ). It is convenient to stress that mappings in ℰ (ℝ2 ) are not necessarily onto. The Poincaré map of a periodic equation in the conditions of Massera’s theorem (uniqueness and global existence for the future) is an example of embedding. A homeomorphism of the plane can be seen as a map h in ℰ (ℝ2 ) which satisfies the additional condition h(ℝ2 ) = ℝ2 and has a continuous inverse.1 In contrast to homeomorphisms, embeddings may not commute with the topological operations of closure and boundary in ℝ2 , denoted by cl(⋅) and 𝜕. A simple example is provided by 1 Later it will be proved that the continuity of the inverse is automatic. https://doi.org/10.1515/9783110551167-003

38 | 3 Free embeddings of the plane the map h(x, y) = (arctan x, y). For the set A = ]−∞, 0[ × ℝ one has π h(cl(A)) = ]− , 0] × ℝ, 2

π cl(h(A)) = [− , 0] × ℝ 2

and h(𝜕A) = {0} × ℝ,

π 𝜕h(A) = {− , 0} × ℝ. 2

This undesirable situation cannot occur if A is bounded. Lemma 3. Given h ∈ ℰ (ℝ2 ), h(cl(A)) = cl(h(A)),

h(𝜕A) = 𝜕h(A),

for every bounded subset A of ℝ2 . Proof. We take a large disk D which is closed and contains A. The restriction of the map to the disk is a homeomorphism between D and h(D). This is so because the map is bijective and continuous and D is compact. The sets D and h(D) are closed in ℝ2 and so the closure of A relative to D coincides with cl(A). The same holds for the closure of h(A) relative to h(D). As the closure commutes with homeomorphisms, we conclude that h(cl(A)) = h(clD (A)) = clh(D) (h(A)) = cl(h(A)). A similar argument works for the boundaries. In this case we assume that the closure of A is contained in the interior of D. Then 𝜕A = cl(ℝ2 \ A) ∩ cl(A) = cl(D \ A) ∩ cl(A). The example above shows that embeddings are not necessarily closed mappings. Next we show that they are always open. Lemma 4. Every map h in ℰ (ℝ2 ) is open. Proof. We must prove that if G is an open subset of ℝ2 , then also h(G) is open. Every open set can be expressed as an union of open and bounded subsets and so we can assume that G is also bounded. As in the previous proof we take a closed disk D containing the closure of G in its interior. It is well known that h defines a homeomorphism from D onto h(D) and, in particular, G is homeomorphic to h(G). The theorem of invariance of the domain2 implies that h(G) is open in the plane. 2 Given G1 and G2 , subsets of ℝd that are homeomorphic, if G1 is open then G2 is also open.

3.1 Topological embeddings | 39

The previous result implies that Ω = h(ℝ2 ) is open and the inverse map h−1 : Ω ⊂ ℝ2 → ℝ2 is continuous. In particular, 2

2

2

2

ℋ(ℝ ) = {h ∈ ℰ (ℝ ) : h(ℝ ) = ℝ }.

Exercise 22. Describe all the subsets of the plane that can be expressed as the image of an embedding. Hint: an open annulus is not in this class. Since h is open, we can deduce that the operation of topological interior, int(⋅), commutes with h in all cases. Lemma 5. Given h ∈ ℰ (ℝ2 ), h(int(A)) = int(h(A)) for every subset A of ℝ2 . Proof. The set Ω = h(ℝ2 ) is open and h is a homeomorphism from ℝ2 onto Ω. Therefore int(h(A)) = intΩ (h(A)) = h(int(A)). Given a Jordan curve Γ ⊂ ℝ2 , the bounded component of ℝ2 \ Γ will be denoted by Ri (Γ). The other component will be Re (Γ). The Schönflies theorem says that the couple (D, Γ) is homeomorphic to (𝔻, 𝕊1 ), where D = Ri (Γ) ∪ Γ,

𝔻 = {z ∈ ℂ : |z| ≤ 1},

𝕊1 = {z ∈ ℂ : |z| = 1}.

Moreover, 𝜕D = Γ and int(D) = Ri (Γ). A subset of the plane that is homeomorphic to 𝔻 will be called a topological disk, or simply a disk. Let us recall the following fact: if two disks in the plane D1 and D2 are such that 𝜕D1 = 𝜕D2 then D1 = D2 . To understand fully this assertion it is useful to observe that it would be false in the sphere. We can interpret Ri (⋅) as an operation acting on planar Jordan curves. The next result shows that Ri commutes with embeddings. Lemma 6. Given h ∈ ℰ (ℝ2 ), h(Ri (Γ)) = Ri (h(Γ)) 2

for every Jordan curve Γ ⊂ ℝ . Proof. First we observe that h defines a homeomorphism from D = Ri (Γ) ∪ Γ onto h(D) = D1 . From Lemma 3 we deduce that the boundary of the disk D1 is 𝜕D1 = h(Γ). Since h(Γ) is a Jordan curve, we have a second disk D2 = Ri (h(Γ)) ∪ h(Γ). The disks D1 and D2 share their boundaries and so they coincide. In particular int(D1 ) = int(D2 ) and the conclusion of the lemma follows from Lemma 5. Exercise 23. Prove that Re (h(Γ)) ∩ Ω = h(Re (Γ)).

40 | 3 Free embeddings of the plane

3.2 Orientation-preserving embeddings A map h ∈ ℰ (ℝ2 ) preserves the orientation if it has degree one; that is, deg(h − q0 , 𝒰 ) = 1, where q0 = h(0) and 𝒰 is any bounded and open neighborhood of the origin. To understand this notion intuitively we consider a Jordan curve Γ around the origin, that is, 0 ∈ Ri (Γ). This implies that q0 ∈ Ri (h(Γ)). The condition on the degree for 𝒰 = Ri (Γ) says that if we parameterize Γ counter-clockwise then h(Γ) will make one positive revolution around q0 . See Figure 3.2.

Figure 3.2: An orientation-preserving map.

The only root of h(x) − q0 = 0 is the origin and so, by excision, it is enough to compute the degree on one particular neighborhood 𝒰 . Also, it is clear that the role of the origin in the above definition can be played by any other point. Indeed, given p⋆ ∈ ℝ2 and q⋆ = h(p⋆ ), we observe that deg(h − q⋆ , B) = deg(h − q0 , B) if B is an open ball which is large enough. To this end we consider an arc α : [0, 1] → ℝ2 joining 0 and p⋆ and define the homotopy H(t, x) = h(x) − h(α(t)). The map H does not vanish on the boundary of any ball B containing the arc α([0, 1]). Exercise 24. Assume that h1 , h2 ∈ ℰ (ℝ2 ) and there exists a non-empty open set G ⊂ ℝ2 such that h1 = h2 in G. Then h2 is orientation-preserving if h1 has this property. The class of orientation-preserving embeddings will be denoted by ℰ⋆ (ℝ2 ) and the prototype is the identity. More generally, we have the inclusion 2 2 Diff∞ ⋆ (ℝ ) ⊂ ℰ⋆ (ℝ ).

3.2 Orientation-preserving embeddings | 41

This is easily proved using the formula for the computation of the degree by the linearization principle. The model of a map which does not preserve the orientation is the symmetry h(x, y) = (x, −y). This is again a consequence of the same formula but it is interesting to visualize it in Figure 3.3.

Figure 3.3: An orientation-reversing map.

In the following exercises we list some properties that can be deduced from Leray’s product theorem (see the appendix). Exercise 25. Given h ∈ ℰ (ℝ2 ) with q = h(0) and 𝒰 an open neighborhood of the origin, prove that deg(h − q, 𝒰 ) = −1 if h is not orientation-preserving. Exercise 26. Prove that if h1 , h2 ∈ ℰ∗ (ℝ2 ) then h1 ∘ h2 ∈ ℰ∗ (ℝ2 ). Exercise 27. Given h ∈ ℰ (ℝ2 ) then h2 ∈ ℰ∗ (ℝ2 ). The Poincaré map of a periodic equation is orientation-preserving. More precisely, assume that X : ℝ × ℝ2 → ℝ2 is continuous, T-periodic in the first variable and such that there is uniqueness for the initial value problem associated to ẋ = X(t, x). In addition we assume that all the solutions are defined in the future and so 𝒟 = ℝ2 . The claim is that the Poincaré map P belongs to ℰ⋆ (ℝ2 ). To prove this we take a disk Δ0 with 0 ∈ int(Δ0 ) and consider the homotopy defined by the flow H : Δ0 × [0, T] → ℝ2 ,

H(p, t) = x(t, p) − x(t, 0).

This is a continuous function and the uniqueness of the Cauchy problem implies that it cannot vanish when the point (p, t) lies in 𝜕Δ0 × [0, T]. Since H0 = id and HT = P − q0 with q0 = P(0), we can apply the homotopy property of the degree to conclude that deg(P − q0 , int(Δ0 )) = deg(id, int(Δ0 )). The set int(Δ0 ) contains the origin and so this degree is 1. For later use it will be convenient to extend the notion of orientation-preserving embedding to maps h : G ⊂ ℝ2 → ℝ2 defined on an open and connected subset of the plane which are continuous and one-to-one. The previous discussions extend to this setting without substantial differences.

42 | 3 Free embeddings of the plane

3.3 Limit sets, trivial dynamics and free embeddings Given h ∈ ℰ (ℝ2 ) and p ∈ ℝ2 , the positive orbit pn = hn (p), n ≥ 0, is well defined. The ω-limit set is defined as Lω (p) = {q ∈ ℝ2 : pσ(n) → q for some sequence of integers σ(n) → +∞}. This set is invariant under h and, if {pn }n≥0 is bounded, it is also non-empty and compact. In general Lω (p) is not connected but it has a related property. Lemma 7. Assume that {pn }n≥0 is bounded and there exist two compact subsets A and B of ℝ2 satisfying Lω (p) = A ∪ B,

A ∩ B = 0,

h(A) ⊂ A.

Then either A = 0 or B = 0. Proof. Assume by contradiction that both sets are non-empty and let Aϵ and Bϵ be open neighborhoods of A and B, respectively, with Aϵ ∩ Bϵ = 0. It is easy to prove that pn ∈ Aϵ ∪ Bϵ ,

for large n.

(3.1)

The neighborhoods Aϵ and Bϵ can be chosen small enough so that if x ∈ Aϵ then h(x) ∈ ̸ Bϵ . Here one uses that A is positively invariant under h and also that h is uniformly continuous on Aϵ . From the assumptions we know that {pn } has to enter infinitely many times in Aϵ and Bϵ . This fact implies the existence of a subsequence pσ(n) lying in Aϵ and such that h(pσ(n) ) ∈ ̸ Aϵ . By construction we know that h(pσ(n) ) cannot belong to Bϵ and this is incompatible with (3.1). We say that h ∈ ℰ (ℝ2 ) has trivial dynamics if Lω (p) ⊂ Fix(h) for every p ∈ ℝ2 . From now on, the set of fixed points of h will be denoted by Fix(h). We observe that translations and contractions are examples of mappings with trivial dynamics. In the former case the limit sets are empty, while in the latter case they only contain the fixed point. Exercise 28. Construct h ∈ ℰ (ℝ2 ) having trivial dynamics and such that Lω (p) is not a singleton for some p. The property of trivial dynamics imposes strong restrictions on the limit set. Proposition 2. Assume that h ∈ ℰ (ℝ2 ) has trivial dynamics and {pn }n≥0 is bounded. Then Lω (p) is a continuum contained in Fix(h). Proof. Let us recall that a continuum is a non-empty, compact and connected set. The proof follows from Lemma 7 if we observe that when Lω (p) is contained in Fix(h) then any subset of Lω (p) is invariant under h.

3.3 Limit sets, trivial dynamics and free embeddings | 43

A map h ∈ ℰ (ℝ2 ) if free if, given any topological disk D ⊂ ℝ2 with h(D) ∩ D = 0, then hn (D) ∩ D = 0

for each n ≥ 2.

Proposition 3. Assume that h ∈ ℰ (ℝ2 ) is free. Then it has trivial dynamics. Proof. We can take an arbitrary point q in ℝ2 \ Fix(h) and prove that it does not belong to Lω (p). As q is not fixed we can find a small disk D with q ∈ int(D) and such that h(D)∩D = 0. Next we prove that the positive orbit {pn }n≥0 visits D at most once. Indeed, if r > s ≥ 0 were integers with hr (p) and hs (p) lying in D, then hr (p) = hr−s (hs (p)) ∈ hr−s (D) and so hr−s (D) ∩ D ≠ 0. This cannot occur since h is free. Once we know that hn (p) eventually remains outside D we conclude that Lω (p)∩int(D) = 0 and so q ∈ ̸ Lω (p). The simplest example of a free embedding is the identity, for if h = id then any disk intersects its image and so the condition for n ≥ 2 is vacuously satisfied. An example of a map which is not free is a rotation. In this case any disk must intersect infinitely many of its iterates. Another example of a non-free embedding is the contraction. This is more subtle as this map has trivial dynamics. Assume for instance h(x) = 21 x and consider the arc γ described in Figure 3.4, where A = (1, 0), B = ( 41 , − 415 ), C = ( 41 , 0). Then h(γ) ∩ γ = 0 and the point C = h2 (A) belongs to h2 (γ) ∩ γ. Finally, we observe that, if ϵ > 0 is small enough, then √

Dϵ = {p ∈ ℝ2 : dist(p, γ) ≤ ϵ} is a disk with h(Dϵ ) ∩ Dϵ = 0 and h2 (Dϵ ) ∩ Dϵ ≠ 0.

Figure 3.4: The arc and the contraction.

In principle one could try to extend the notion of free embedding to other dimensions. This can be done after replacing the topological disks of the plane by sets in ℝd that are homeomorphic to the closed ball {p ∈ ℝd : ‖p‖ ≤ 1}. Brown noticed that h = id

44 | 3 Free embeddings of the plane is the only free homeomorphism in dimension d ≥ 3. Following his ideas it is easy to show that a translation is not free in ℝ3 . Consider the translation T(x, y, z) = (x, y + 1, z) ̂ ∪ PQ ̂ ∪ QR, ̂ where O = (0, 0, 0), P = (0, 0, 1), Q = and the piecewise linear arc Γ = OP (1, 0, 0), R = (0, 2, 0). We observe that T(Γ)∩Γ = 0 but T 2 (Γ)∩Γ = {R}. See Figure 3.5. Now we can inflate Γ slightly so that it becomes a topological ball Γϵ satisfying T(Γϵ )∩Γϵ = 0 and T 2 (Γϵ ) ∩ Γϵ ≠ 0.

Figure 3.5: The arc and the translation.

At this point the reader may wonder whether there are free embeddings in the plane different from the identity. Exercise 29. Prove that h ∈ ℰ (ℝ) is free if and only if h is increasing. Exercise 30. A map h ∈ ℰ (ℝ2 ) is called locally free if for each x ∈ ℝ2 there exists a disk Dx such that x ∈ int(Dx ) and hn (Dx ) ∩ Dx = 0 for each n ≥ 1. Prove that a locally free map has trivial dynamics. Extend this notion to higher dimensions and construct a locally free map in ℝ3 different from the identity.

3.4 Brown’s degree condition We present a result allowing one to check that a map is free just by computing certain degrees. Theorem 6. Assume that h ∈ ℰ⋆ (ℝ2 ) and deg(id − h, Ri (Γ)) ≠ 1, for every Jordan curve Γ ⊂ ℝ2 \ Fix(h). Then h is free.

(3.2)

3.4 Brown’s degree condition

| 45

This result, due to M. Brown, is somewhat unusual. As free embeddings have trivial dynamics, it shows an instance where one can extract dynamical information from degree theory. Brown’s proof was presented for homeomorphisms and we shall adapt it to embeddings. This will be the main task of the rest of the chapter. Let us first discuss some consequences of this theorem. First we observe that (3.2) is satisfied if h is fixed point free. This means that Fix(h) = 0 and, in this case, deg(id − h, Ri (Γ)) = 0 for every Jordan curve Γ ⊂ ℝ2 . In particular translations are free, for they are orientation-preserving and fixed point free. Now we present two examples of free embeddings having fixed points, h1 (x, y) = (x + sin y, y),

1 h2 (x, y) = ( x, 2y). 2

It is easy to prove that h1 and h2 are in ℰ⋆ (ℝ2 ). For h1 the set of fixed points is the family of parallel lines y = nπ, n ∈ ℤ. For any Jordan curve Γ with Γ ∩ Fix(h1 ) = 0, we also have Ri (Γ) ∩ Fix(h1 ) = 0. This implies that (3.2) holds, the degree vanishing always. For h2 the only fixed point is the origin and −1 if 0 ∈ Ri (Γ), deg(id − h2 , Ri (Γ)) = { 0 if 0 ∈ Re (Γ). These degrees are easily computed since the map id − h2 is linear and has negative determinant. It is interesting to observe that the condition h ∈ ℰ⋆ (ℝ) is essential for the validity of the theorem. To show this consider the map h(x, y) = (x + 1, −y). As Fix(h) = 0 the condition (3.2) holds. We cannot apply Theorem 6 in this case and we are going to prove that h is not free. First observe that the orbit starting at p = (0, 1) is pn = (n, (−1)n ). Let D be a topological disk containing p0 and p2 and lying in {y > 0}. Then h(D) is contained in {y < 0} and so D and h(D) do not intersect. Since p2 ∈ D∩h2 (D) the map is not free.3 Many interesting results on planar dynamics can be obtained as corollaries of Theorem 6. We present a result going back to Brouwer. 3 It can be proved that every free embedding is orientation-preserving. The intuition behind this fact will be explained in Section 3.8.

46 | 3 Free embeddings of the plane Corollary 1. Assume that h ∈ ℰ⋆ (ℝ2 ) and, for some p ∈ ℝ2 , lim inf ‖hn (p)‖ < ∞. n→∞

Then h has at least one fixed point. Proof. First we observe that the assumption on the orbit starting at p says that Lω (p) is non-empty. Assume by a contradiction argument that Fix(h) = 0. Then h would be free and we could apply Proposition 3 to deduce that h should have trivial dynamics. Thus Lω (p) ⊂ Fix(h) and this is the searched contradiction. Exercise 31. Assume that h ∈ ℰ⋆ (ℝ2 ) has a recurrent point4 which is not fixed. Prove the existence of ϵ > 0 such that if ĥ ∈ ℰ⋆ (ℝ2 ) and ̂ ‖h(p) − h(p)‖ ≤ϵ

for every p ∈ ℝ2 ,

then ĥ has at least one fixed point.

3.5 Translation arcs and Brouwer’s lemma We shall work with oriented arcs. These are subsets of the plane which are homeomorphic to a compact interval and such that the end points are ordered. An oriented arc ̂ will be denoted by α = p q, where p and q are the end points. 2 ̂ Given h ∈ ℰ (ℝ ), α = p q is a translation arc if h(p) = q and h(α \ {q}) ∩ (α \ {q}) = 0. This definition implies that translations arcs do not contain fixed points. This is clear for the points in α different from q, but if q were fixed then h(p) = q = h(q) would imply that p = q, which is absurd since α is an arc.

Figure 3.6: A translation arc.

Figure 3.6 shows a typical translation arc, but it can also happen that h(q) lies on α\{q}. This is the case for the map h = −id and the arc α(θ) = (cos θ, sin θ), θ ∈ [0, π]. The 4 A point p is recurrent if p ∈ Lω (p).

3.5 Translation arcs and Brouwer’s lemma

| 47

points p = (1, 0) and q = (−1, 0) compose a 2-periodic orbit. An example where h(q) is a point in α \ {p, q} is given by the map h(x, y) = (x + 1, −y) and the polygonal arc passing through the points p0 = (0, 0), r = (0, 1), s = (2, 1), p2 = (2, 0) and p1 = (1, 0). ̂ Exercise 32. Find a map h ∈ ℰ (ℝ2 ) and an arc α = p q such that h(p) = q, h(α \ {p}) ∩ (α \ {p}) = 0 and α is not a translation arc. The iterates hn (α) of a translation arc can have different behaviors. For the translation h(x, y) = (x + 1, y) and the arc α = [0, 1] × {0}, we have hn (α) ∩ α = 0 if n ≥ 2. On the contrary, for the rotation of 90 degrees (counter-clockwise) and the arc α(θ) = (cos θ, sin θ), θ ∈ [0, π2 ], one has h3 (α) ∩ α ≠ 0. For orientation-preserving maps this second situation can only occur if h has fixed points. Theorem 7 (Brouwer’s lemma). Assume that h ∈ ℰ⋆ (ℝ2 ) and α is a translation arc with hn (α) ∩ α ≠ 0

for some n ≥ 2.

Then there exists a Jordan curve Γ ⊂ ℝ2 \ Fix(h) such that deg(id − h, Ri (Γ)) = 1. To get a better understanding of this result, it is interesting to analyze an example where it fails because h is not orientation-preserving. The map h(x, y) = (2x, −y) has a unique fixed point at the origin, while the other points on the y axis are 2-periodic. We observe that deg(id − h, Ri (Γ)) is either −1 or 0. The arc α(θ) = (cos θ, sin θ), θ ∈ [ π2 , 3π ], 2 n is a translation arc satisfying h (α) ∩ α ≠ 0 for each n ≥ 2. Exercise 33. Construct a similar example where deg(id − h, Ri (Γ)) = 0 whenever it is defined. Exercise 34. Assume that h ∈ ℰ∗ (ℝ2 ) has no fixed points and let α be a translation arc. Then α ∩ hn (α) = 0 for each n ≥ 2. Brouwer’s lemma will be proved in Chapter 5. By now we present an intuitive argument in the simplest situation. Towards a proof of Theorem 7: a first attempt. Assume that the arc α is such that p is a 2-periodic point. Then Γ = α ∪ h(α) is a Jordan curve. As h is one-to-one, the arcs h(α) and h2 (α) will only intersect at p and q. The arcs α and h2 (α) can intersect at many points. Assume that Γ has the counter-clockwise orientation, there are two possible cases: (i) h(Γ) has a counter-clockwise orientation. See Figure 3.7. (ii) h(Γ) has a clockwise orientation. See Figure 3.8. The second case is discarded since h is orientation-preserving. Once we know that we can concentrate in case (i), we will compute the degree of id − h on Ri (Γ). To this

48 | 3 Free embeddings of the plane

Figure 3.7: Case (i).

Figure 3.8: Case (ii).

end we compute the variation of the argument of the vector v⃗ = x − h(x) when x goes around Γ. First we compute the variation when x goes through α, denoted by Δ(α). At x = p the vector v⃗ is p − q and, as x moves through α, it changes continuously until x = q is reached, where it becomes q − p. In the process the argument can have many variations but at the end the angle has been increased in 180 degrees and so Δ(α) = π. Similarly, we compute the variation of the argument of v⃗ through h(α). Now we travel from q − p to p − q and obtain Δ(h(α)) = π. Finally, using the formula expressing the degree in terms of the argument function, we conclude that deg(id − h, Ri (Γ)) =

1 [Δ(α) + Δ(h(α))] = 1. 2π

3.6 Existence of translation arcs There are maps in the plane that do not admit translation arcs. Clearly this is the case for the identity, but there are less obvious examples. The symmetry h(x, y) = (x, −y) has no translation arcs, since any arc connecting p to h(p) must cross the x axis. This is precisely the fixed point set Fix(h) and we know that translation arcs do not contain fixed points. The following result shows that these examples can be regarded as exceptional. Proposition 4. Assume that h ∈ ℰ (ℝ2 ) and D is a topological disk with D and h(D) lying on the same component of ℝ2 \ Fix(h). In addition assume that h(D) ∩ D = 0. ̂ Then, given points w1 , . . . , wn ∈ D, there exists a translation arc α = p q with w1 , . . . , wn ∈ α̇ := α \ {p, q}.

3.6 Existence of translation arcs | 49

With respect to the applicability of this result, it is convenient to observe that we can construct a translation arc as soon as we can find a point w which is not fixed and can be connected to h(w) by a path not having fixed points. Indeed, small disks around w will not intersect their images and so the above proposition can be applied. In this way we find a translation arc with w ∈ α.̇ It would be wrong to think that the ? obtained by gluing the sub-arcs w ? of h(α), is always ̂ arc β = wh(w), q of α and qh(w) a translation arc. As an example we can consider the homeomorphism of the plane (expressed in polar coordinates), h:

θ1 = φ(θ),

r1 = r,

where φ is continuous, strictly increasing, and it satisfies φ(θ + 2π) = φ(θ) + 2π,

θ ∈ ℝ.

If we impose the requirements φ(0) = π,

φ(π) = 2π,

π 3π φ( ) = , 2 2

φ(

3π 5π )> , 2 2

then the arc α : r = 1, θ ∈ [0, π], is a translation arc with end points p = (1, 0), ? given by r = 1, q = (−1, 0). However, if we choose w = (0, 1) ∈ α,̇ the arc β = wh(w), ], is not a translation arc. θ ∈ [ π2 , 3π 2 The rest of this section will be devoted to the proof of Proposition 4. We start with a definition: a family of disks {Dt }t∈[0,1] will be called admissible if Dt = ⋂ Ds s>t

for t ∈ [0, 1[ and int(Dt ) = ⋃ Ds s 0, we consider the family of disks Et = {z ∈ ℝ2 : ‖z‖ ≤ 1 + m(t)} ∪ ([0, 1 + t] × [−m(t), m(t)]). This is an admissible family and the same can be said for {Dt } with Dt = ϕ(Et ). See Figure 3.10. Let us fix ϵ > 0 such that 2

𝒰 = {z ∈ ℝ : dist(z, D ∪ α) < ϵ}

Figure 3.10: Modification of the disk.

3.6 Existence of translation arcs | 51

is contained in G. We take a sequence of functions mn : [0, 1] → ℝ in the above conditions and such that mn (1) → 0 as n → +∞. The corresponding disk E1 will converge to 𝔻 ∪ 𝔸 in the Hausdorff distance. In consequence D1 converges to D ∪ α. Selecting n so that mn (1) is small enough, we can guarantee that D1 ⊂ 𝒰 ⊂ G. The smaller disks Dt will also be contained in G. By construction, D = D0 and q is a point in D1 ∩ Δ. Exercise 35. Construct a subset S ⊂ ℝ2 which is homeomorphic to 𝕊1 ∪ 𝔸 and such that it is not possible to find ϕ ∈ ℋ(ℝ2 ) with ϕ(𝕊1 ∪ 𝔸) = S. Lemma 9. Assume that {Dt } and {Δt } are admissible families of disks with D0 ∩ Δ0 = 0,

D1 ∩ Δ1 ≠ 0.

Then, for some τ ∈ ]0, 1], int(Dτ ) ∩ int(Δτ ) = 0,

𝜕Dτ ∩ 𝜕Δτ ≠ 0.

Proof. Define τ = sup{t ≥ 0 : Dt ∩ Δt = 0}. We first prove that Dτ ∩ Δτ ≠ 0. This is clear from the assumption if τ = 1. If τ < 1 we use the first condition for admissibility to deduce that Dτ ∩ Δτ = ⋂ (Ds ∩ Δs ). s>τ

Cantor’s lemma on decreasing sequences of compact sets can applied to conclude that Dτ ∩ Δτ is non-empty. In particular, τ > 0. Both families of disks are increasing and so the second condition for admissibility leads to int(Dτ ) ∩ int(Δτ ) = ⋃ (Ds ∩ Δs ). s 0 as in Lemma 9, where Δt = h(Dt ). Take z ∈ 𝜕Dτ ∩ 𝜕h(Dτ ). See Figure 3.11. We use Lemma 3 to find w ∈ 𝜕Dτ such that z = h(w). Again Lemma 3 implies that h(z) ∈ 𝜕h(Dτ ). We observe that z is not a fixed point, since it belongs to Dτ , a subset of ℝ2 \ Fix(h). Thus, z ≠ w and z ≠ h(z), although sometimes it can happen that w and h(z) coincide. The points w1 , . . . , wn lie in D0 , contained in the interior of ̂z with w1 , . . . , wn ∈ α̇ ⊂ int(Dτ ). In view of Lemma 4, Dτ . We select an arc α = w (α \ {z}) ∩ h(α \ {z}) ⊂ [int(Dτ ) ∪ {w}] ∩ [int(h(Dτ )) ∪ {z}] = 0. This says that α is a translation arc.

52 | 3 Free embeddings of the plane

Figure 3.11: The touching disks.

3.7 The complement of the fixed point set In order to apply Proposition 4 to a map h one needs to know that the disks D and h(D) lie in the same component of ℝ2 \ Fix(h). The purpose of this section is to show that this condition is automatically satisfied if h is orientation-preserving. This is a consequence of the following result. Proposition 5. Assume that h ∈ ℰ⋆ (ℝ2 ) and let {Ui }i∈I be the family of connected components of ℝ2 \ Fix(h), where I ⊂ ℕ. Then h(Ui ) ⊂ Ui for each i ∈ I. When h is a homeomorphism the image of a connected component of ℝ2 \ Fix(h) is also a connected component. In such case the conclusion of the above proposition can be refined to h(Ui ) = Ui . This is a consequence of a result by Brown and Kister dealing with homeomorphisms in arbitrary dimension. The proof of Proposition 5 will combine some ideas taken from the elegant proof by Brown and Kister and a classical result on planar involutions due to Kerékjártó. Given a topological space X, an involution of X is a continuous map g : X → X, g ≠ id, with g 2 = g ∘ g = id. This definition implies that g is its own inverse and so it is a homeomorphism. For X = 𝔻 or ℝ2 , the prototypes of involutions are g1 (x, y) = (−x, −y) and g2 (x, y) = (x, −y). There is a result going back to Brouwer and Kerékjártó saying that any involution g in 𝔻 or ℝ2 is conjugate to g1 or g2 . This means that g = ϕ∘gi ∘ϕ−1 for some ϕ ∈ ℋ(X), where X = 𝔻 or ℝ2 and i = 1 or 2. This result is also valid in any topological disk, now ϕ is understood as a homeomorphism between 𝔻 and D. To distinguish between g1 and g2 we observe that only g1 is orientation-preserving. The notion of orientation-preserving embeddings was introduced in Section 3.2 and it clearly extends to embeddings with local domain, say a closed disk or an open and connected subset of the plane. In particular the class ℋ⋆ (X) has an obvious meaning for X = 𝔻2 or ℝ2 . The invariance of the degree under changes of variable implies that ℋ⋆ (X) is invariant under conjugacy and so any involution in ℋ⋆ (X) is conjugate to g = −id.

3.7 The complement of the fixed point set | 53

Let W be an open and connected subset of the plane and let g : W ⊂ ℝ2 → ℝ2 be a continuous and one-to-one map. We say that g is a weak involution if g ≠ id and g(g(z)) = z

for each z ∈ g −1 (W).

Weak involutions are not always involutions in the strict sense. The simplest instance being a map with g(W) ∩ W = 0. The map g1 has a unique fixed point at the origin and this implies that any involution of the disk which is orientation-preserving must have a unique fixed point. This is not the case for weak involutions. Next we present one of these maps having two fixed points. Consider the set 𝒲 composed of the union of two disjoint disks D1 and D2 together with a thin tube joining them. The open and connected set W ⊂ ℝ2 will be the interior of 𝒲 . The disks have centers at the points p1 and p2 . The transformation g : W ⊂ ℝ2 → ℝ2 maps W homeomorphically onto g(W) and satisfies g −1 (W) = int(D1 ) ∪ int(D2 ). Moreover, the restriction of g to int(Di ) is the involution z 󳨃→ 2pi − z. See Figure 3.12. It is clear that g is an orientation-preserving weak involution and Fix(g) = {p1 , p2 }.

Figure 3.12: A weak involution with two fixed points.

Lemma 10. Given an orientation-preserving weak involution g : W ⊂ ℝ2 → ℝ2 , the set of fixed points Fix(g) is discrete. Before proving this lemma we recall a well known fact about the intersection of topological disks. Given two disks D1 and D2 in the plane, let C denote any connected component of int(D1 ) ∩ int(D2 ) and Δ = cl(C). Then Δ is a topological disk with int(Δ) = C. Exercise 36. Let 𝔹 = {z ∈ ℝ3 : ‖z‖ ≤ 1} denote the standard closed ball in three dimensions. Find sets B1 , B2 ⊂ ℝ3 , homeomorphic to 𝔹, such that C = int(B1 ) ∩ int(B2 ) is connected and Δ = cl(C) is not homeomorphic to 𝔹. Proof of Lemma 10. We first prove that any fixed point z⋆ ∈ Fix(g) must be in one of the following categories: – z⋆ is isolated in Fix(g); – there exists a neighborhood U of z⋆ such that g|U = id.

54 | 3 Free embeddings of the plane To check this we shall assume that z⋆ is not isolated and prove that the second alternative holds. We start by observing that the continuity of g allows one to find a disk D with z⋆ ∈ int(D)

and D ∪ g(D) ⊂ W.

In particular D ⊂ g −1 (W) and so g(g(z)) = z for each z ∈ D. This implies that the set X = D ∩ g(D) is mapped onto itself and the restriction of g, denoted as gX : X → X, is an involution in the strict sense. Since D and g(D) are homeomorphic, g(D) is also a disk with z⋆ = g(z⋆ ) ∈ int(g(D)). The last assertion can be justified by observing that the boundary of the disk is a topological invariant. Let C denote the connected component of int(D) ∩ int(g(D)) containing z⋆ . This component is invariant under g and so is the disk Δ = cl(C). The restriction gΔ : Δ → Δ is again an involution. Since gΔ is orientation-preserving, Kerékjártó’s theory says that either gΔ = id or gΔ is conjugate to g1 : 𝔻 → 𝔻, g1 = −id. We are assuming that z⋆ is not isolated as a fixed point and so the conjugacy with g1 is excluded. We conclude that gΔ = id so that the second alternative holds. To complete the proof we show that the set Y = {z ∈ W : g|U = id

for some neighborhood U of z}

is empty. This set is clearly open and the previous alternative for fixed points implies that it is also closed (relative to W). By assumption g is not the identity and so Y cannot be the whole W. This implies that Y = 0 and so all fixed points are isolated. Proof of Proposition 5. We argue by contradiction and assume that, for some j ∈ I, the image h(Uj ) is not contained in Uj . The set h(Uj ) ⊂ ℝ2 \ Fix(h) is connected and so it must be contained in some other component, say h(Uj ) ⊂ Uk with k ∈ I, k ≠ j. In particular, Uj ∩ h(Uj ) = 0.

(3.3)

We know from Lemma 4 that Ω = h(ℝ2 ) is open and h−1 : Ω ⊂ ℝ2 → ℝ2 is continuous. Define the new map 2

2

g:Ω⊂ℝ →ℝ ,

h(z) if z ∈ Ω ∩ Uj , { { { −1 g(z) = {h (z) if z ∈ h(Uj ), { { otherwise. {z

The condition (3.3) guarantees that g is well defined. The definition of g implies that Fix(h) ⊂ Fix(g).

(3.4)

The key to arrive at the searched contradiction is that g is an orientation-preserving weak involution. Once this fact is accepted we notice that, since Uj is a proper subset

3.7 The complement of the fixed point set | 55

of ℝ2 , there exists a point z⋆ ∈ 𝜕Uj . This point must be fixed under h and, from h(Uj ) ⊂ Uk , it must satisfy z⋆ ∈ 𝜕Uj ∩ 𝜕Uk .

(3.5)

Lemma 10 implies that z⋆ is isolated in Fix(g) and, from (3.4), we can deduce that it is also isolated in Fix(h). We can find a disk D with z⋆ ∈ int(D) and (D \ {z⋆ }) ∩ Fix(h) = 0. The set D \ {z⋆ } is connected, therefore it must be contained in one of the components of ℝ2 \ Fix(h). This is not compatible with (3.5). The proof will be complete if we check that g is a weak involution. This is done in five steps. Step 1. g is one-to-one. Define the sets R1 = Ω ∩ Uj ,

R2 = h(Uj ),

R3 = Ω \ (R1 ∪ R2 ).

This is a partition of Ω and g(R1 ) ⊂ R2 ,

g(R2 ) ∩ Ω ⊂ R1 ,

g(R3 ) = R3 .

The restrictions g|Ri are one-to-one for each i = 1, 2, 3. These facts are easily combined to prove that g is one-to-one on the whole Ω. Step 2. g is continuous. From ℝ2 = ⋃i∈I Ui ∪ Fix(h), it follows that Ω = ⋃ h(Ui ) ∪ Fix(h). i∈I

Also, we observe that, for any two indices i, l ∈ I, i ≠ l, then h(Ui ) ∩ h(Ul ) = 0 and either h(Ui ) ⊂ Ul or h(Ui ) ∩ Ul = 0. Assume now that we are given a sequence in {zn } lying in Ω and converging to some point z ∈ Ω. We must prove that g(zn ) converges to g(z). This is clear if z ∈ Fix(h), since the three sequences h(zn ), h−1 (zn ) and zn converge to z. Assume now that z is in h(Ui ) for some i. The set h(Ui ) is open and the sequence zn eventually enters into this set. If i = j then g(zn ) = h−1 (zn ) for large n and this sequence converges to g(z) = h−1 (z). The same argument applies (with h−1 replaced by h) if i ≠ j but h(Ui ) ⊂ Uj . Finally, if i ≠ j and h(Ui ) ∩ Uj = 0, the points zn and z are fixed under g. Step 3. g(g(z)) = z for each z ∈ g −1 (Ω). This is straightforward. Step 4. g is orientation-preserving. The maps h and g coincide on the open set Ω ∩ Uj and we claim that this set is non-empty. Then, given q ∈ h(Ω ∩ Uj ), deg(g − q, Ω ∩ Uj ) = deg(h − q, Ω ∩ Uj ) = 1,

56 | 3 Free embeddings of the plane where we have used the fact that h is orientation-preserving and the excision property of the degree.5 It remains to prove that Ω ∩ Uj is non-empty. Indeed any point z⋆ ∈ 𝜕Uj is fixed under h and therefore it belongs to Ω. Any small disk D around z⋆ will be contained inside the open set Ω. Then D ∩ Uj has to be a non-empty subset of Ω. Step 5. g ≠ id. Note that g(z) ≠ z if z ∈ Ω ∩ Uj .

3.8 More on free embeddings and a proof of Brown’s theorem We shall combine the results obtained in Sections 3.5, 3.6 and 3.7 to prove Theorem 6. After this, we present some additional properties of free embeddings. These properties will not be used later and the discussion will be more informal. Proof of Theorem 6. We shall assume that h is not free and find a Jordan curve Γ ⊂ ℝ2 \ Fix(h) with deg(id − h, Ri (Γ)) = 1. In view of Brouwer’s lemma (Theorem 7), it will be sufficient to find a translation arc α such that hn (α) ∩ α ≠ 0 for some n ≥ 2. If h is not free, we can find a disk D with h(D) ∩ D = 0 and hn (D) ∩ D ≠ 0 for some n ≥ 2. As h is orientation-preserving, we can apply Proposition 5 to deduce that D and h(D) lie in the same component of ℝ2 \ Fix(h). Let z be a point in hn (D) ∩ D and w ∈ D with hn (w) = z. We apply Proposition 4 and find a translation arc α with w, z ∈ α.̇ This implies that z belongs to hn (α) ∩ α. In this way we have found the searched translation arc. The proof is complete but it is worth to notice that w and z could coincide. This case is included in the previous reasoning, the only difference being that Proposition 4 is applied to one point w1 = w = z instead of two points w1 = z, w2 = w. Next we shall discuss why free embeddings are always orientation-preserving, as was announced in Section 3.4. We shall assume that h is a free embedding not in ℰ⋆ (ℝ2 ) and we shall be lead to a contradiction. From Proposition 3 we know that h has trivial dynamics. In particular h has no periodic points and Fix(h) = Fix(h2 ). It follows from Exercise 27 that the map h2 = h ∘ h is in ℰ⋆ (ℝ2 ) and Proposition 5 implies that the components of ℝ2 \ Fix(h) are mapped into themselves by h2 ; that is, h2 (Ui ) ⊂ Ui . We distinguish two cases. Case 1: h(U1 ) ⊂ U2 for two different components. Since h2 (U1 ) ⊂ U1 we can take a point z ∈ U1 and construct a disk D ⊂ U1 with 2 z, h (z) ∈ D. See Figure 3.13. It is clear that h(D) ∩ D = 0 because h(D) ⊂ U2 . On the 5 This argument is related to Exercise 24.

3.8 More on free embeddings and a proof of Brown’s theorem

| 57

Figure 3.13: Case 1.

other hand we know that the point h2 (z) lies in h2 (D) ∩ D and then h is not free, a contradiction. Case 2: h(Ui ) ⊂ Ui for each i. We apply Proposition 4 and find a translation arc α = p̂ 0 p1 . At this point we are using the fact that h is not the identity. Define γ = ⋃4k=0 hk (α). This second case now bifurcates in two sub-cases. The notation pk = hk (p0 ) is employed. Sub-case 2.a: γ = p̂ 0 p5 is an arc. We construct a disk Δ with p0 , p5 ∈ 𝜕Δ, γ̇ ⊂ int(Δ) and such that Δ \ γ is split in two components Δ+ and Δ− . See Figure 3.14.

Figure 3.14: The disk Δ in Case 2a.

Now we take a new arc β with end points at p3 and p1 and β̇ ⊂ Δ− ∩ h−1 (Δ). We are going to prove that β can be chosen so that h(β)̇ ∩ γ = 0.

(3.6)

To get this property we assume that the arc β is very close to h(α)∪h2 (α), the continuity of h implies that also h(β) is very close to h2 (α) ∪ h3 (α). Then h(β) and α will be disjoint. An arc under these conditions will satisfy (3.6), for otherwise there should exist a point ξ in h(β)̇ ∩ γ and, in particular, ξ ∈ h(β)̇ ∩ hr (α) for some r = 1, 2, 3, 4. In consequence

58 | 3 Free embeddings of the plane we can find η ∈ β̇ and σ ∈ α such that ξ = h(η) = hr (σ). This implies η = hr−1 (σ) ∈ β̇ ∩ γ, a contradiction. Once we know that (3.6) holds, we observe that the connected set h(β)̇ is contained in Δ \ γ and therefore either h(β)̇ ⊂ Δ+ or h(β)̇ ⊂ Δ− . See Figures 3.15 and 3.16.

Figure 3.15: h(β)̇ ⊂ Δ+ , Case 2a.

Figure 3.16: h(β)̇ ⊂ Δ− , Case 2a.

Consider the Jordan curve Γ = h(α) ∪ h2 (α) ∪ β with the clockwise orientation. Then h(Γ) = h2 (α) ∪ h3 (α) ∪ h(β) has the same orientation if h(β)̇ ⊂ Δ− and the counterclockwise orientation if h(β)̇ ⊂ Δ+ . We conclude that the second situation holds because h is orientation-reversing.6 Once we know that β̇ ⊂ Δ− and h(β)̇ ⊂ Δ+ , we conclude that h(β)∩β = 0. We can inflate β slightly so that it becomes a disk with h(D)∩D = 0. Then h is not free because p3 ∈ h2 (β) ∩ β ⊂ h2 (D) ∩ D. Sub-case 2.b: γ = p̂ 0 p5 is not an arc. We can find points w, z ∈ α \ {p1 } such that hr (w) = z for some r ≥ 2. The sub-arc ̂z, satisfies h(β) ∩ β = 0 and hr (β) ∩ β ≠ 0. of α with end points at w and z, say β = w It is now easy to prove that h is not free by enlarging β so that it becomes a disk. See Figure 3.17. Notice that the case w = z is not excluded in the previous argument. To finish this section we notice that the family of free embeddings is closed by passage to the limit. More precisely, assume that {hk } is a sequence in ℰ (ℝ2 ) and h ∈ ℰ (ℝ2 ) is such that hk (z) → h(z) for each z ∈ ℝ2 , 6 This is just an intuitive argument!

3.9 Two fixed point theorems | 59

Figure 3.17: Case 2b.

and this convergence is uniform in each compact subset of the plane. If hk is free for each k, then h is free. To justify this statement we proceed by contradiction and assume that h is not free. Then there should exist a disk D and n ≥ 2 such that h(D) ∩ D = 0 and hn (D) ∩ D ≠ 0. We can enlarge D a little, say D ⊂ Δ, in such a way that the new disk Δ satisfies h(Δ) ∩ Δ = 0 and hn (int(Δ)) ∩ int(Δ) ≠ 0. Let z and w be points in int(Δ) with hn (z) = w. For large k the point hnk (z) belongs to int(Δ) and so wk = hnk (z) is a point in hnk (int(Δ)) ∩ int(Δ). Also, hk (Δ) ∪ Δ = 0. This shows that hk is not free.

3.9 Two fixed point theorems In this section we prove two classical results on fixed points of planar maps. At first sight these results seem unrelated, but we shall discover that they have a common feature: they can be proved using the theory of free embeddings.

3.9.1 Area-preserving maps of the open disk Consider the open disk Δ = {p ∈ ℝ2 : ‖p‖ < 1} and the class of orientation-preserving homeomorphisms of the disk, denoted by ℋ∗ (Δ). There are maps h : Δ → Δ in this class without fixed points. As an example consider a flow {ϕt }t∈ℝ defined on the closed disk cl(Δ) with the phase portrait of Figure 3.18. The points on the boundary E± = (±1, 0) are equilibria and the remaining orbits are heteroclinic connections from E− to E+ . If we fix a time τ > 0, the map h = ϕτ is in ℋ∗ (Δ) and has no fixed points in Δ. This situation is impossible if h is area-preserving. We say that h is area-preserving if λ(h(G)) = λ(G)

60 | 3 Free embeddings of the plane

Figure 3.18: A flow on the disk.

for each open set G ⊂ Δ. Here λ is the Lebesgue measure on the disk. Theorem 8. Assume that h ∈ ℋ∗ (Δ) is area-preserving. Then it has a fixed point. As far as I know this result appeared for the first time as a final remark in a paper by Montgomery of 1945. There was no proof but it was said that it is a consequence of Brouwer’s translation theorem. Before going into the details of the proof, we present some examples showing that the assumptions are sharp. Example 1. We consider a new flow {ψt }t∈ℝ of the closed disk which is area-preserving and has the phase portrait given by Figure 3.19.

Figure 3.19: An area-preserving flow.

To be sure that such flow exists we employ the Hamiltonian formalism and consider the system of differential equations ẋ = x2 + 3y2 − 1,

ẏ = −2xy

associated to the Hamiltonian function H(x, y) = (x 2 + y2 − 1)y,

(x, y) ∈ ℝ2 .

A well known theorem in Classical Mechanics guarantees that the map ψt is areapreserving for each t. The energy level H = 0 is the union of the real line ℝ × {0} and the unit circle 𝕊1 . In consequence Δ is invariant under the flow and it is not hard

3.9 Two fixed point theorems | 61

to show that the phase portrait corresponds to the previous picture. In particular there are two centers at C± = (0, ± √13 ) and a symmetry with respect to the x axis. If we fix τ > 0 then the map h1 = ψτ is under the conditions of the theorem and the points C+ and C− are fixed. Exercise 37. Is the identity Fix(hI ) = {C+ , C− } valid for every τ > 0? Example 2. Let us now define h2 : Δ → Δ, h2 = S ∘ hI where S : ℝ2 → ℝ2 , S(x, y) = (x, −y), is the symmetry with respect to the x axis. The map h2 is certainly area-preserving but the previous result is not applicable because it is orientationreversing. Actually, it has no fixed points. This example shows that the preservation of the orientation is essential. Example 3. Now we show that the theorem cannot be extended to higher dimensions. Let C = Δ × ]−σ, σ[ be a solid cylinder with coordinates (x, y) ∈ Δ, z ∈ ]−σ, σ[. Define h3 : C → C,

h3 (x, y, z) = (h2 (x, y), −z).

This map is a homeomorphism-preserving volume and orientation but it has no fixed points. It is possible to transport the example to the open ball Δ3 = {(x, y, z) ∈ ℝ3 : x2 + y2 + z 2 < 1}. Given a volume-preserving homeomorphism ψ : C → Δ3 , define h∗3 = ψ ∘ h3 ∘ ψ−1 . The number σ must be chosen appropriately, so that C and Δ3 have the same volume. Exercise 38. Construct similar examples in arbitrary dimensions. In many results connecting measure and topology it is possible to replace the class of measure-preserving maps by a larger class of maps defined only in topological terms. We present such a class of maps. Given a homeomorphism h ∈ ℋ(Δ) we say that h is conservative (or non-dissipative) if for each open and non-empty set G ⊂ Δ there exists an integer n = n(G) ≥ 1 such that hn (G) ∩ G ≠ 0. A simple example of non-conservative map is the homeomorphism h(p) = ‖p‖p,

p ∈ Δ.

The origin attracts radially all orbits in the open disk and, as soon as r22 ≤ r1 , the annulus G = {p ∈ Δ : r1 < ‖p‖ < r2 } satisfies hn (G) ∩ G = 0 for each n ≥ 1. Examples of conservative maps can be produced by means of the following result. Lemma 11. Assume that μ is a finite measure on Δ with the additional property: open sets are measurable and have positive measure. Let h ∈ ℋ(Δ) be a measure preserving map, that is, μ(h(G)) = μ(G), for each open set G ⊂ Δ. Then h is conservative.

62 | 3 Free embeddings of the plane Proof. Given a non-empty open set G ⊂ Δ we observe that all iterates hn (G) have the same positive measure, say μ(hn (G)) = m > 0. This implies that the sets G, h(G), h2 (G), . . . , hn (G), . . . cannot be pairwise disjoint. To prove this claim we proceed by contradiction. We apply the σ-additivity of the measure to deduce that, if these sets were disjoint, then ∑ μ(hn (G)) = μ( ⋃ hn (G)) ≤ μ(Δ) < ∞,

n≥0

n≥0

which is impossible if m > 0. Let r > s ≥ 0 be integers such that hr (G) ∩ hs (G) ≠ 0, then hr−s (G) ∩ G ≠ 0 and we can take n(G) = r − s. We are ready to present an extension of Theorem 8. Theorem 9. Assume that h ∈ ℋ∗ (Δ) is a conservative homeomorphism. Then either h is the identity on Δ or there exists a Jordan curve Γ ⊂ Δ such that Fix(h) ∩ Γ = 0 and deg(id − h, Ri (Γ)) = 1. Even for area-preserving maps, this result gives more information than the previous Theorem 8. Now we know that there is a region inside Δ with fixed point index equal to one. The case h = idΔ is exceptional. Proof. We will employ a topological version of Poincaré’s recurrence theorem due to Oxtoby. For a conservative homeomorphism h ∈ ℋ(Δ) and any open set G ⊂ Δ, all points of G except a set of first category are recurrent with respect to G. This means that, for most points p ∈ G, there is an infinite number of integers n ≥ 1 such that hn (p) ∈ G. The expression “for most points” is understood in the sense of category (small set = set of first category). Assume that h ∈ ℋ∗ (Δ) is not the identity. Then the open set Δ\Fix(h) is non-empty and we consider the family of connected components {Ui }i∈I . Since h is orientationpreserving we know by Proposition 5 that h(Ui ) ⊂ Ui for each i. As h is a homeomorphism, also h(Ui ) must be a component, therefore each Ui is invariant under h. Let us fix some Ui = h(Ui ). We can find a small disk D ⊂ Ui with h(D) ∩ D = 0. This is possible because there are no fixed points in Ui . From the initial discussion we know that most points in int(D) are recurrent with respect to this set. In particular there exist a point p ∈ int(D) and an integer n ≥ 2 such that hn (p) ∈ int(D). The invariance of Ui implies that h(D) is also contained in this component, we can invoke Proposition 4 with w1 = p and w2 = hn (p) to find a translation arc α with p, hn (p) ∈ α.̇ The point hn (p) lies simultaneously in α and hn (α) and we are under the conditions of Brouwer’s lemma. Notice that Proposition 5, Proposition 4 and Brouwer’s lemma are concerned with maps defined on the whole plane. Since ℝ2 and Δ are homeomorphic, they are easily translated to this setting.

3.9 Two fixed point theorems | 63

3.9.2 Fixed points on invariant continua In a series of papers published in the period 1945-49, Cartwright and Littlewood analyzed the dynamics of the forced Van der Pol oscillator. Among other results, they proved that the attractor is sometimes a continuum with complicated topology. This important observation was their initial motivation for the study of fixed points on invariant continua. By a continuum we understand a non-empty, compact and connected subset of the plane. We say that the continuum K ⊂ ℝ2 does not separate the plane if ℝ2 \ K is connected. A simple strategy to look for examples consists in the interpretation of capital letters of the alphabet as planar continua. The letters H, T or X do not separate the plane. A more sophisticated example is the baroque 𝒮 , defined as the union of two disjoint closed segments and an open curve winding around them. See Figure 3.20.

Figure 3.20: A continuum in the plane.

The following result deals with general continua. Theorem 10. Assume that h ∈ ℋ∗ (ℝ2 ) and let K ⊂ ℝ2 be a continuum which does not separate the plane. In addition assume that K is invariant under h, h(K) = K. Then h has a fixed point lying in K. The conclusion of this theorem fails for some continua separating the plane. This is the case for K = 𝕊1 . This set is invariant under any rotation h around the origin of the plane, but the only fixed point is the origin and does not belong to K. Exercise 39. Construct a continuum K ⊂ ℝ2 separating the plane and such that any homeomorphism h : K → K has a fixed point. Exercise 40. Prove that the theorem still holds when K is positively invariant, h(K) ⊂ K. The original proof of the theorem by Cartwright and Littlewood was rather long and employed a sophisticated theory of planar continua (prime ends). We will present

64 | 3 Free embeddings of the plane a proof due to M. Brown combining free embeddings and covering spaces. First we recall the definition and some basic properties of covering spaces. Assume that X and X̃ are metric spaces which are arcwise connected and locally arcwise connected.7 In addition assume that ℘ : X̃ → X is a continuous map with the following property: for each point x ∈ X there exists an arcwise connected neighborhood U such that ℘−1 (U) is decomposed in a family of arcwise connected components {Uα }α∈A such that ℘ : Uα → U is a homeomorphism for each α ∈ A. It is always assumed that ℘−1 (U) is non-empty. A map with these properties is called a covering map. Sometimes we will say that X̃ is a covering space of X. The basic example is X = 𝕊1 = {z ∈ ℂ : |z| = 1},

X̃ = ℝ,

℘(θ) = eiθ .

Given a point z = eiθ we can define U as an arc centered at z with length α < 2π. Then ℘−1 (U) = ⋃n∈ℤ Un , Un = ]θ + 2πn − α2 , θ + 2πn + α2 [. See Figure 3.21.

Figure 3.21: The most famous covering map.

A related example is X = ℂ \ {0}, X̃ = ℂ, ℘(z) = eiz . The link between these examples is the following property. Given a subset A ⊂ X which is arcwise connected and locally arcwise connected, we select an arcwise connected component à of ℘−1 (A). Then the restricted map ℘ : à → A is a covering map. It is well known that every open and connected subset of the plane G ⊂ ℝ2 has a covering map of the type ℘ : ℝ2 → G. This is a delicate fact that follows from the theory of Riemann surfaces. When G is simply connected all covering maps from ℝ2 to G are homeomorphisms. On the contrary, ℘ is not one-to-one when G has holes. Exercise 41. Find an open and connected set G ⊂ ℝ3 such that there is no covering map ℘ : ℝ3 → G. From our point of view, the main property of covering spaces will be the lifting of homeomorphisms. Given a covering map ℘ : X̃ → X with X̃ simply connected, assume that h : X → X is a homeomorphism and select points x0 , y0 ∈ X, x̃0 , ỹ0 ∈ X̃ with h(x0 ) = y0 , ℘(x̃0 ) = x0 , ℘(ỹ0 ) = y0 . Then there exists a homeomorphism h̃ : X̃ → X̃ 7 This is always the case for an open and connected subset of the Euclidean space.

3.9 Two fixed point theorems | 65

satisfying h(̃ x̃0 ) = ỹ0 and ℘ ∘ h̃ = h ∘ ℘. As an example consider the counter-clockwise ̃ rotation of 90 degrees in 𝕊1 . If we select x0 = 1, y0 = i, x̃0 = 2π, ỹ0 = π2 , then h(θ) = 3π θ− 2 . Before turning to the proof of the theorem we present a preliminary result. Lemma 12. Assume that K ⊂ ℝ2 is a continuum which does not separate the plane and G is an open subset of ℝ2 with K ⊂ G. Then there exists a topological disk D satisfying K ⊂ int(D) ⊂ D ⊂ G. See Figure 3.22.

Figure 3.22: A continuum K inside a disk D.

Proof. It will be convenient to work on the Riemann sphere 𝕊2 = ℝ2 ∪ {∞}. By assumption we know that Ω = 𝕊2 \K is connected and it is not hard to prove that it is also simply connected. The set Ω is homeomorphic to the open disk and we can find a homeomorphism ℛ : Δ → Ω with ℛ(0) = ∞. For each r ∈ ]0, 1[ we define Cr = {z ∈ ℂ : |z| = r} and γr = ℛ(Cr ). See Figure 3.23.

Figure 3.23: A neighborhood of K.

It is easy to prove that the following equivalence holds: |zn | → 1

⇐⇒

dist(ℛ(zn ), K) → 0,

(3.7)

for any sequence {zn } with zn ∈ Δ. From this we deduce that K ⊂ Ri (γr ). This is proved by contradiction. Otherwise K should be contained in the unbounded component of ℝ2 \ γr and, for any sequence approaching K, say {pn } with pn ∈ ̸ K, dist(pn , K) → 0, the sequence zn = ℛ−1 (pn ) should lie in the disk |z| < r. Consider the decreasing family of disks Dr = γr ∪ Ri (γr ),

0 < r < 1.

66 | 3 Free embeddings of the plane We claim that Dr accumulates on K as r → 1. This means that limr→0 h(Dr , K) = 0 where h(Dr , K) = max{dist(p, K) : p ∈ Dr }. Since K ⊂ Dr , this claim is a consequence of the equivalence (3.7). Now it is easy to complete the proof. For small ϵ > 0, the set Kϵ = {p ∈ ℝ2 : dist(p, K) ≤ ϵ} is contained in G. The set Kϵ is a neighborhood of K and so Dr is contained in Kϵ for r close to 1. Exercise 42. Given a continuum K ⊂ ℝ2 , prove that every connected component of 𝕊2 \ K is simply connected. Proof of Theorem 10. We proceed by contradiction and assume that Fix(h) ∩ K = 0. Then K is contained in a connected component U of ℝ2 \ Fix(h). From Lemma 12 we deduce that there exists a disk D satisfying K ⊂ int(D) ⊂ D ⊂ U. Let us take a covering map ℘ : ℝ2 → U. This is possible because U is an open and connected subset of the plane. The set ℘−1 (int(D)) is composed by a family of arcwise connected components {Δα }α∈A . Each restricted map ℘α : Δα → int(D) is also a covering map. Since int(D) is simply connected, ℘α is indeed a homeomorphism. Define the sets Kα = ℘−1 α (K). They are continua lying in Δα . Moreover, ℘−1 (K) = ⋃ Kα α∈A

with Kα1 ∩ Kα2 = 0 if α1 ≠ α2 . This means that {Kα }α∈A is the family of connected components of ℘−1 (K). Notice that they are not necessarily arcwise connected components.8 In Figure 3.24 we consider a hypothetical situation of a homeomorphism having a unique fixed point at the origin, Fix(h) = {0}, and an arc of circumference as the invariant continuum. This arc should be lifted to a family of segments. After completing the proof, it will be clear that the following situation is impossible: U = ℂ \ {0},

℘(z) = eiz ,

K = {eiθ : |θ| ≤

π }, 2

Kn = [−

π π + 2πn, + 2πn]. 2 2

8 For the continuum presented in Figure 3.20, each Kα is connected but it has three arcwise connected components.

3.9 Two fixed point theorems | 67

Figure 3.24: The family of continua ℘−1 (K).

Let us select some β ∈ A and a point x0 ∈ K. Since K is invariant we know that y0 = h(x0 ) ∈ K and we pick x̃0 , ỹ0 ∈ Kβ with ℘(x̃0 ) = x0 and ℘(ỹ0 ) = y0 . The homeomorphism h maps U onto itself and we can lift h : U → U to the covering space ℝ2 . Consider the homeomorphism h̃ : ℝ2 → ℝ2 with ℘ ∘ h̃ = h ∘ ℘ and h(̃ x̃0 ) = ỹ0 . From the product theorem by Leray (see the appendix), we know that h̃ is orientation-preserving, that is, h̃ ∈ ℋ∗ (ℝ2 ). We claim that ̃ )⊂K . h(K β β To prove this inclusion we first observe that ̃ )) = h(℘(K )) = h(K) = K. ℘(h(K β β ̃ ) is contained in ℘−1 (K). The connected component of ℘−1 (K) containing the Then h(K β ̃ ) is precisely K . This is a consequence of the choice of the lifting, since continuum h(K β β ̃ ). ỹ0 lies simultaneously in Kβ and h(K β ̃ ) ⊂ K holds, we deduce that any forward Once we know that the inclusion h(K β β ̃ orbit of h starting at Kβ must remain in this set for the future. Then Corollary 1 can be ̃ We observe that x = ℘(x)̃ ∈ U applied to deduce that h̃ has a fixed point, say x̃ = h(̃ x). is a fixed point of h, because ̃ = ℘(h(̃ x)) ̃ = ℘(x)̃ = x. h(x) = h(℘(x)) This is not compatible with the initial definition of the set U, contained in the complement of Fix(h). The previous Theorem is also valid for orientation-preserving embeddings h : G ⊂ ℝ → ℝ2 where G is an open set containing K. From Lemma 12 we know that it is possible to find a topological disk D with K ⊂ int(D) ⊂ D ⊂ G. Let D̂ be a smaller disk with K ⊂ int(D)̂ ⊂ D̂ ⊂ int(D). See Figure 3.25. 2

68 | 3 Free embeddings of the plane

Figure 3.25: Modification and extension of h.

Then we can construct homeomorphisms ψ1 : D → ℝ2 and ψ2 : h(D) → ℝ2 such that ̂ The map ĥ = ψ2 ∘ h ∘ ψ−1 belongs to ℋ∗ (ℝ2 ) and h = ĥ ψ1 = id on D̂ and ψ2 = id on h(D). 1 on D.̂ The previous Theorem can be applied to h.̂

3.10 Proof of Massera’s theorem The classical version, Theorem 3, is a rather direct consequence of Corollary 1. Let us consider the Poincaré map associated to (2.1). We know from Section 3.2 and the assumption 𝒟 = ℝ2 that P is in the class ℰ∗ (ℝ2 ). Given a solution x(t) bounded in the future, we can assume that it is well defined on [0, ∞[, for otherwise we would consider the solution x(t + nT), where n is a sufficiently large integer. This solution will produce an orbit of P with lim supn→∞ ‖P n (p)‖ < ∞. Then P has a fixed point corresponding to the initial condition of a T-periodic solution. The proof of Theorem 4 requires more work. Proof of Theorem 4. Assume by contradiction that there exists a solution x(t) defined on [0, +∞[ with lim inf ‖x(t)‖ < ∞. t→+∞

Then we can select a sequence of positive numbers tn → +∞ and a point q ∈ ℝ2 such that limn→+∞ x(tn ) = q. Let us decompose each instant tn in the form tn = τn + σ(n)T,

τn ∈ [0, T[, σ(n) ∈ ℤ.

After extracting a subsequence we can assume that τn converges to some τ ∈ [0, T]. Also, σ(n) → +∞ as n → +∞. Let y(t) be the solution of (2.1) with initial condition y(τ) = q. The assumption of global existence implies that y(t) is well defined on [τ, +∞[. The continuous dependence with respect to initial conditions implies that ̂ ≤δ there exists δ > 0 such that if z(t) is another solution with |τ − τ|̂ + ‖y(τ) − z(τ)‖ then z(t) is well defined in [τ, τ + T] and ‖y(s) − z(s)‖ ≤ 1

if τ ≤ s ≤ τ + T.

Define zn (s) = x(s + σ(n)T). This is a solution defined on [−σ(n)T, +∞[ and such that zn (τn ) = x(tn ) converges to q. Then, for large n and s ∈ [τ, τ+T], ‖y(s)−zn (s)‖ ≤ 1. In particular, ‖y(T) − zn (T)‖ ≤ 1. Let us now consider the Poincaré map P associated to (2.1).

3.10 Proof of Massera’s theorem | 69

We know that P belongs to ℰ∗ (ℝ2 ). At least for large n, the point zn (T) = P σ(n)+1 (x(0)) lies in the ball of radius 1 and center at y(T). In consequence, lim inf ‖P n (x(0))‖ ≤ ‖y(T)‖ + 1 < ∞, n→+∞

and we can apply Corollary 1 to conclude that P has a fixed point. This is not possible since we are assuming that equation (2.1) has no T-periodic solutions. The less standard result Theorem 5 is a consequence of Brown’s degree condition. Proof of Theorem 5. Let φ(t) be a periodic solution of (2.1) which is not of period T. Then p = φ(0) is not a fixed point of P but it is recurrent, meaning that p ∈ Lω (p). To prove the recurrence we assume that τ > 0 is a period of φ and distinguish two cases: (i) τ is commensurable with T; that is, Tτ = − sr ∈ ℚ. Then P r (p) = φ(rT) = φ(−sτ) = φ(0) = p, and p is a periodic point. In particular, P nr (p) = p → p as n → +∞. (ii) τ is not commensurable with T; that is, Tτ ∈ ̸ ℚ. In this case there exist sequences of integer numbers sn and rn with rn → +∞ and −sn τ + rn T → 0. Then P rn (p) = φ(rn T) = φ(rn T − sn τ) → φ(0) = p. We can now invoke Proposition 3 to deduce that P is not free. In view of Theorem 6 we deduce that Brown’s degree condition does not hold and we find a Jordan curve Γ ⊂ ℝ2 with Fix(P) ∩ Γ = 0 and deg(id − P, Ri (Γ)) = 1. Consider the disk D = Ri (Γ) ∪ Γ, the stability of the degree under small perturbations implies the existence of a number ν > 0 such that if Q : D → ℝ2 is a continuous map satisfying ‖P(p) − Q(p)‖ < ν

if p ∈ Γ,

(3.8)

then Fix(Q) ∩ Γ = 0 and deg(id − Q, Ri (Γ)) = 1. Let us select a large R > 2ν satisfying ‖x(t, p)‖ ≤ R2 if t ∈ [0, T] and p ∈ D. Let X ∗ : ℝ × ℝ2 → ℝ2 be a continuous and T-periodic vector field. It is not restrictive to assume that there is uniqueness for the initial value problem associated to this perturbed equation ẋ = X ∗ (t, x). The solution satisfying x(0) = p will be denoted by y(t, p). The compact set K is chosen as the closed ball centered at the origin with radius R. By continuous dependence it is possible to find δ > 0 such that, for any vector field X ∗ in the previous conditions and such that ‖X − X ∗ ‖ ≤ δ on ℝ × K, the following properties hold: – For each p ∈ D the solution y(t, p) is defined on [0, T]; – Given t ∈ [0, T] and p ∈ D, ‖x(t, p) − y(t, p)‖ < ν. Let P ∗ be the Poincaré map associated to the perturbed equation, P ∗ (p) = y(T, p). The map Q = P ∗ is continuous in D and satisfies (3.8). In consequence Fix(P ∗ ) ∩ Γ = 0 and deg(id − P ∗ , Ri (Γ)) = 1.

70 | 3 Free embeddings of the plane This implies the existence of a T-periodic solution x∗ (t) of ẋ = X ∗ (t, x) with p∗ = x∗ (0) ∈ D. Then, for each t ∈ [0, T], 󵄩 R 󵄩 󵄩 󵄩 ‖x∗ (t)‖ ≤ 󵄩󵄩󵄩x(t, p∗ )󵄩󵄩󵄩 + 󵄩󵄩󵄩x(t, p∗ ) − y(t, p∗ )󵄩󵄩󵄩 ≤ + ν < R. 2 The solution x∗ (t) remains in K forever.

3.11 Bibliographical remarks As regards Section 3.1, the theorem of invariance of the domain is valid in any topological manifold (see [42]). The reader is referred to [87] for more information on the Jordan–Schönflies theorem. As regards Section 3.2, the orientation of a topological manifold is usually defined in terms of the non-trivial local homology group (see [42]). Orientation-preserving embeddings can be characterized as the class of embeddings inducing the identity on this homology group. The definition in terms of the degree is intuitive and perhaps more accessible to analysts. As regards Section 3.3, as far as I know, Proposition 2 first appeared as an auxiliary result in a paper by Hale and Raugel (see Lemma 2.7 in [53]). It is also proved in [29]. The notion of free homeomorphism was introduced by Brown in [24]. In that paper it is observed that free homeomorphisms have trivial dynamics. Free homeomorphisms without fixed points appeared in the older paper by Andrea [6]. The argument employed to prove that the identity is not free in three dimensions is inspired by the proof of Lemma 6.3 in [24]. The notion of locally free map was introduced in [24]. As regards Section 3.4, the version of Theorem 6 for homeomorphisms is Theorem 5.7 in [24]. Brown says that this was his starting point for the study of free homeomorphisms. For analytic homeomorphisms, Theorem 6 can be sharpened: the omega limit set is either empty or a singleton. Therefore, every orbit converges to a fixed point or to infinity (see [99]). The origin of Corollary 1 is in the paper by Brouwer [21]. As regards Section 3.5, translation arcs were introduced by Brouwer in [21]. Some of the authors who used this notion later are v. Kerékjàrtó, Terasaka, Sperner and Scorza Dragoni. Reference [118] has a very complete list of references. Brouwer’s paper [21] is dedicated to the plane translation theorem. This theorem says that any orientation-preserving embedding without fixed points is (in a semi-local sense) conjugate to a translation. This result has been extended recently in several directions [51, 18, 71, 46]. The traditional version of Brouwer’s lemma is given in Exercise 34. This is Satz 1 in [21]. See also [6, 23]. The version of Brouwer’s lemma we have stated is Theorem 5.7 in [24]. Very modestly, Brown refers to Brouwer’s paper for a proof. The paper by Fathi [44] is also concerned with Brouwer’s lemma. As regards Section 3.6, the existence of translation arcs was already discussed by Brouwer. The proof we present follows classical arguments. It must be noticed that

3.11 Bibliographical remarks | 71

there are some differences with respect to Lemma 4.1 in [24]. The prescribed points w1 , . . . , wn are not end points of the translation arc. The variant of the Schönflies Theorem employed in the proof of Lemma 8 follows from a theorem due to Homma [59]. A triod is a set homeomorphic to the letter y. According to Homma’s result, the homeomorphism exists if all triods have the same orientation. The union of a circumference and a segment with one common point has triods whose orientation depends on whether the segment is inside or outside the disk. Some related results can be found in the books [93, 67]. As regards Section 3.7, the statement and the proof of Proposition 5 are inspired the result of Brown and Kister in [25]. The result in [25] is concerned with homeomorphisms defined on an orientable manifold of arbitrary dimension. In contrast, Proposition 5 deals with embeddings of the plane. Several exclusive properties of the plane have been used in the proof and I do not know if the same conclusion can be obtained for embeddings of ℝd with d ≥ 3. The topological classification of planar involutions is due to Kérékjartó, with a later contribution of Eilenberg. See [35] for more details. The result on the intersection of two topological disks in the plane can be obtained as a consequence of the converse of Jordan’s theorem. See the book by Newman [93], where it is called the Kéréjartó theorem. A complementary view of the result is found in the book by Pommerenke [109], Proposition 2.13. As regards Section 3.8, the informal discussion of the connection between free embeddings and the property of preservation of orientation is adapted from the proof of Theorem 4.9 in [24]. As regards Section 3.9.1, as mentioned in the main text, Theorem 8 was stated by Montgomery in [88]. Proofs were presented in the papers by Andrea [6] and Bourgin [19]. This result is attributed to Montgomery in [6]. The question of the possible extension of the theorem to higher dimension was posed in [19]. This reference contains a counter-example for the disk Δ135 . A high dimension was required because the construction relied on the existence of periodic maps without fixed points in Euclidean spaces of certain dimensions. Later, Asimov constructed in [9] a counter-example in Δ3 . In this case the construction was based on the Hopf fibration. Finally, Alpern constructed in [4] a counter-example in Δ2 for an orientation-reversing map. Alpern’s construction employs some results on general measure-preserving transformations due to Oxtoby and Ulam. Examples 2 and 3 provide an elementary method to construct counter-examples. Area-preserving maps can be seen as symplectic diffeomorphisms in one degree of freedom (two dimensions) and it makes sense to ask if Theorem 8 could have an extension to canonical maps in more degrees of freedom. It was shown by Morrison in [89] that also in this setting there are counter-examples. Using the Hopf fibration, he constructed an interesting example of a symplectic diffeomorphism of Δ4 without fixed points. The notion of conservative homeomorphism is taken from the excellent book [103], although the terminology has been changed (conservative instead of non-dissipative). The not so well-known topological version of Poincaré’s

72 | 3 Free embeddings of the plane Recurrence Theorem in [103] has inspired the formulation of Theorem 9. A previous version of this result appeared in [98]. As regards Section 3.9.2, the theorem of Cartwright and Littlewood appeared for the first time in the beautiful paper of [31]. This paper contains many new ideas on how to employ the theory of prime ends in the study of planar dynamics (see [3, 104, 100, 61] for more recent developments). Later, Hamilton observed in [54] that it is possible to prove Cartwright–Littlewood’s theorem using Brouwer’s theory. The proof sketched by Hamilton depends upon delicate properties of the intersection of a Jordan curve and its image under the homeomorphism. The proof we have presented is due to Brown [22]. Hamilton’s paper has three pages and has the title “A short proof of the Cartwright Littlewood fixed point theorem”, Brown’s paper has only one page but the title is slightly longer (“A short short proof of the Cartwright–Littlewood theorem”). Bell extended the C-L theorem to orientation-reversing homeomorphisms in [11] and, in some cases, there is a second fixed point. This was shown by Kuperberg in [65]. The C-L fixed point theorem has an intrinsic topological interest but it has found applications in differential equations, see in particular [12]. For the general theory of covering spaces I have followed the book [84]. Every open and connected subset of the plane can be interpreted as a Riemann surface and so its universal cover (X̃ simply connected) is also a Riemann surface. According to the uniformization theorem all simply connected Riemann manifolds are conformal to the disk, the plane or the sphere (see [120]). The sphere is excluded because it is compact and so the universal covering space has to be homeomorphic to the whole plane. As regards Section 3.10, the results by Brown in [23] and [24] cannot be directly applied to prove Massera’s theorem. In fact Brown worked with homeomorphisms and the Poincaré map P is just an embedding. This fact was observed in [38] and lead Murthy to present in [92] some discussions of the extension of Brown’s techniques to embeddings.

4 Index of stable fixed points and periodic solutions The notion of stability in the Lyapunov sense can be defined for solutions of discrete and continuous systems. We will explore some topological consequences. Given an isolated fixed point, the associated index is defined in terms of the topological degree. We are interested in the possible values of this index when the fixed point is stable. It will be shown that the answer to this question depends upon the dimension of the phase space. In dimension d = 2 this index must be +1 and this fact has interesting consequences. The final part of the chapter will be devoted to applications in population dynamics and the theory of Hamiltonian systems with two degrees of freedom. The norm of uniform convergence is denoted by ‖⋅‖∞ . Given a continuous function f : [0, ∞[ → ℝd , ‖f ‖∞ = sup ‖f (t)‖. t≥0

Note that ‖f ‖∞ = +∞ whenever f is unbounded.

4.1 Lyapunov’s stability Consider the system of differential equations ẋ = X(t, x),

x ∈ G,

(4.1)

where G is a connected and open subset of ℝd and the vector field X : ℝ × G → ℝd is continuous and periodic in time, X(t + T, x) = X(t, x) if (t, x) ∈ ℝ × G, for some fixed T > 0. In contrast to the setting of Chapter 1, now the vector field is not necessarily defined on the whole space ℝ × ℝd . We still assume the uniqueness for the initial value problem associated to (4.1). Again we employ the notation x(t, p) for the solution with x(0) = p ∈ G. It is defined on the maximal interval ]α, ω[. A solution x(t, p) is called stable if there exists a neighborhood 𝒰 of p such that ω(q) = +∞ for any q ∈ 𝒰 and lim ‖x(⋅, q) − x(⋅, p)‖∞ = 0.

q→p

The traditional theorems on continuous dependence are concerned with a compact interval of time and hold for all solutions. Stability is a more exclusive property that can be viewed as a continuous dependence on the unbounded interval [0, ∞[. The initial instant t0 = 0 has been distinguished in the previous definition but nothing changes if the interval [0, ∞[ is replaced by any [t0 , +∞[ with t0 > α(p). This is a consequence of the property of continuous dependence on compact intervals. https://doi.org/10.1515/9783110551167-004

74 | 4 Index of stable fixed points and periodic solutions We will be mainly interested in the stability of periodic solutions, but sometimes non-periodic solutions can be stable. This is the case for the equation ü + 2u = sin t and the solution u(t) = sin t + sin √2t. After the change of variables x = (u, u)̇ ∗ , we obtain a system of the type (4.1) with G = ℝ2 , T = 2π and p = (0, 1+ √2)∗ . The difference of any two solutions solves the corresponding homogeneous linear system and this leads to the identity 1 1 2 2 (x (t, q) − x2 (t, p)) + (x1 (t, q) − x1 (t, p)) = (q2 − p2 )2 + (q1 − p1 )2 , 2 2 2 implying that 1 ‖x(⋅, q) − x(⋅, p)‖2∞ ≤ ‖q − p‖2 . 2 The definition of stability is usually concerned with the future but stability for the past can also be defined, just replacing [0, +∞[ by ]−∞, 0]. In the previous example there is perpetual stability (future and past). Exercise 43. Construct an example of a non-periodic solution that is stable for the future but not for the past. Exercise 44. Consider equation (4.1) and assume that φn (t) and φ(t) are non-constant T periodic solutions with non-commensurable periods Tn and T ( Tn ∈ ̸ ℚ). Prove that if φn (0) → φ(0) then φ(t) is unstable. Exercise 45. Prove that u ≡ 0 is the only stable solution of ü + u3 = 0. As in Chapter 1 we can consider the Poincaré map P : 𝒟 ⊂ G → G,

P(p) = x(T, p)

where 𝒟 = {p ∈ G : ω(p) > T}. To enlarge the dictionary periodic equations/discrete dynamics we introduce the notion of stability of fixed points. Assume that X is a metric space with distance d. Given an open set 𝒟 ⊂ X and a map M : 𝒟 ⊂ X → X, we consider the difference equation xn+1 = M(xn ),

xn ∈ 𝒟.

(4.2)

We say that a solution is complete if it is defined for every positive integer n ∈ ℕ. The complete solution {xn } is stable if there exists a neighborhood 𝒰 of x0 such that for each y0 ∈ 𝒰 the corresponding solution {yn } is complete and sup d(xn , yn ) → 0 n≥0

as y0 → x0 .

4.1 Lyapunov’s stability | 75

A fixed point of x∗ = M(x∗ ) produces a constant and complete solution. From the previous definition we say that x∗ is stable if there exists a neighborhood 𝒰 of x∗ such that for each x0 ∈ 𝒰 the corresponding solution {xn } is complete and sup d(x∗ , xn ) → 0

as x0 → x∗ .

n≥0

Assume now that φ(t) = x(t, p∗ ) is a T-periodic solution of (4.1) and φ(0) = p∗ is the corresponding fixed point of the Poincaré map. With X = G and M = P we have the equivalence φ(t) is a stable solution of (4.1)

⇐⇒

p∗ is a stable fixed point of P.

The implication (⇒) is a direct consequence of the definitions. The proof of the converse (⇐) is not so direct. We know that p∗ is stable and we must prove that ‖x(⋅, p)− φ‖∞ → 0 as p → p∗ . First we select a closed ball B centered at p∗ and contained in 𝒰 . By continuous dependence the map (t, p) ∈ [0, T] × B 󳨃→ x(t, p) ∈ G is uniformly continuous. Hence we can find a modulus of continuity; that is, an increasing function ω = ω(ξ ) with limξ →0+ ω(ξ ) = 0 such that ‖x(t, p) − x(t, q)‖ ≤ ω(‖p − q‖)

if t ∈ [0, T], p, q ∈ B.

When t ∈ [nT, (n + 1)T], x(t, p) = x(t − nT, pn ) and ‖x(t, p) − φ(t)‖ = ‖x(t − nT, pn ) − x(t − nT, p∗ )‖ ≤ ω(‖pn − p∗ ‖). At this point we must use the stability of p∗ to make sure that all the iterates pn remain in B when ‖p0 − p∗ ‖ is small. Summing up, ‖x(⋅, p) − φ‖∞ ≤ ω(sup ‖pn − p∗ ‖). n≥0

Since p∗ is stable, the quantity supn≥0 ‖pn −p∗ ‖ tends to zero as p0 = p converges to p∗ . To finish this section we discuss some connections between the concept of stability and the existence of certain invariant sets. It is convenient to initiate the discussion in an abstract setting. We consider equation (4.2) on a metric space. The closure of a set A ⊂ X is denoted by cl(A) and the diameter is defined by diam(A) = sup{d(x, y) : x, y ∈ A}. Proposition 6. Assume that the map M is open; that is, M(G) is open if G ⊂ 𝒟 is open. Then a fixed point x∗ = M(x∗ ) is stable if and only if there exists a sequence {𝒰k }k≥1 of open neighborhoods of x∗ satisfying the properties:

76 | 4 Index of stable fixed points and periodic solutions –

𝒰k is strongly decreasing,

cl(𝒰k+1 ) ⊂ 𝒰k ; –

𝒰k shrinks towards the fixed point,

diam(𝒰k ) → 0 as k → ∞; –

𝒰k is positively invariant,

M(𝒰k ) ⊂ 𝒰k . Proof. We assume that x∗ is stable and construct the family of neighborhoods. The other implication is left to the reader. Fix R1 ∈ ]0, 1[ and let B1 = B(x∗ , R1 ) be the open ball centered at x∗ of radius R1 . The stability of x∗ implies that there exists a number r1 > 0 with r1 < R1 such that all the solutions of (4.2) starting at β1 = B(x∗ , r1 ) are complete and remain in B1 . Define n

𝒰1 = ⋃ M (β1 ). n≥0

We observe that 𝒰1 is an open neighborhood of x∗ with β1 ⊂ 𝒰1 ⊂ B1 and M(𝒰1 ) ⊂ 𝒰1 . The process can be repeated inductively to obtain sequences {Rk } and {rk } with Rk+1 < min{rk , k1 } and balls Bk = B(x∗ , Rk ) and βk = B(x∗ , rk ). The corresponding family {𝒰k } satisfies all the conditions. In the previous result it is not necessary to assume continuity of M but this map must be open. This is illustrated by the following example. Let X be the half-line [0, ∞[. The map M : 𝒟 = [0, 21 [ → X is constructed by linear interpolation from the conditions M(

1 1 1 ) = M( )= , 2k 2k + 1 2k

k = 1, 2, . . .

See Figure 4.1. The closed intervals Ik = [0, 2k1 ] are invariant under M and this implies that x∗ is stable. However, it is not possible to find a family {𝒰k } under the conditions of Proposition 6. Otherwise we could consider the connected component of 𝒰k containing the origin. This component must be of the type [0, αk [ with αk > 0. The set M([0, αk [) is connected and contained in 𝒰k . Since 0 ∈ M([0, αk [) we deduce that M([0, αk [) ⊂ [0, αk [. This is impossible because x ∈ M([0, x[) for each x > 0. Let us go back to the general setting and assume that the map M is one-to-one, open and continuous. Then equation (4.2) can be also solved for the past, via the inverse map M −1 : 𝒟̂ ⊂ X → X with 𝒟̂ = M(𝒟). The fixed point is called perpetually stable if it is stable for M and M −1 . The characterization given by Proposition 6 can

4.1 Lyapunov’s stability | 77

Figure 4.1: The function M.

be adapted to perpetually stable fixed points, now the sets 𝒰n are defined by 𝒰n = n ⋃+∞ n=−∞ M (βn ) and they satisfy M(𝒰n ) = 𝒰n . When we go back to the differential equation (4.1), the above results can be applied to the Poincaré map. Given a stable T-periodic solution φ(t), we can find open sets 𝒰k ⊂ G with φ(0) ∈ 𝒰k and P(𝒰k ) ⊂ 𝒰k . The set in ℝd × ℝ, 𝒱k = {(x(t, p), t) ∈ G × [0, ∞[ : p ∈ 𝒰k }

is well defined and homeomorphic to 𝒰k × [0, ∞[. The graph of solutions starting at 𝒰k (for t0 = 0) remains in 𝒱k for the future. When the stability is perpetual, P(𝒰k ) = 𝒰k and hence P(𝜕𝒰k ) = 𝜕𝒰k , because 𝒰k lies inside a compact set contained in 𝒟. A simple situation occurs when the dimension d = 2 and 𝒰k ∪ 𝜕𝒰k is homeomorphic to a disk. Then the boundary of 𝒱k relative to ℝ2 ×[0, ∞[ can be thought of as an invariant

cylinder or, after identifying t = 0 and t = T, as an invariant torus. See Figure 4.2.

Figure 4.2: The invariant cylinder.

78 | 4 Index of stable fixed points and periodic solutions

4.2 The index of a stable equilibrium In this section we consider an autonomous system of arbitrary dimension, ẋ = X(x),

x ∈ G ⊂ ℝd ,

(4.3)

where G is an open set and the vector field X : G ⊂ ℝd → ℝd is continuous. We also assume that there is uniqueness for the initial value problem associated to (4.3). As usual, the solution with x(0) = p is denoted by x(t, p). A constant solution x(t, p∗ ) ≡ p∗ with p∗ ∈ G is associated to each zero of X(x), X(p∗ ) = 0. Sometimes p∗ is called an equilibrium. We say that the equilibrium p∗ is isolated if there exists an open ball B centered at p∗ such that X does not vanish on cl(B) \ {p∗ }. In this case the index of the vector field X at p∗ can be defined as the degree of X at B, deg(X, B). The properties of the degree imply that B can be replaced by any open and bounded neighborhood 𝒰 of p∗ satisfying cl(𝒰 ) ⊂ G and X(p) ≠ 0 for each p ∈ cl(𝒰 ) \ {p∗ }. We would like to discuss the possible values of the index at an isolated and stable equilibrium. This is a delicate question whose answer changes with the dimension. First we consider a special situation where it is easier to give a complete answer. We say that the equilibrium p∗ is an attractor for (4.3) if there exists a neighborhood 𝒱 of p∗ such that 𝒱 ⊂ G, ω(p) = +∞ for each p ∈ 𝒱 and x(t, p) → p∗

as t → +∞.

Note that an attractor is always isolated. An equilibrium is asymptotically stable if it is an attractor and a stable solution for (4.3). Proposition 7. Assume that p∗ is an asymptotically stable equilibrium. Then the index of the vector field X at p∗ is (−1)d . Proof. It is divided in three steps. First step. Assume that 𝒱 is as in the definition of attractor and let K be a compact subset of 𝒱 . Then lim x(t, p) = p∗

t→+∞

uniformly in p ∈ K. Let us fix ϵ > 0. We must find τ > 0 such that ‖x(t, p) − p∗ ‖ < ϵ

if p ∈ K and t ≥ τ.

(4.4)

First we use the stability of p∗ to find δ > 0 such that ‖x(t, q) − p∗ ‖ < ϵ

if t ≥ 0 and ‖q − p∗ ‖ < δ.

(4.5)

4.2 The index of a stable equilibrium | 79

Given Q ∈ K we observe that x(t, Q) is attracted by p∗ . In consequence there exists some τ = τ(Q) > 0 such that ‖x(t, Q) − p∗ ‖
0 such that deg(Ft , B) = 1

if t ≥ τ.

The identity FT (p) = 0 implies that x(T, p) = p and therefore x(t, p) is T-periodic. Since x(t, p) is attracted by p∗ this implies p = p∗ . Let us now compute the degree. In view of the first step we can find τ > 0 such that if p ∈ cl(B) then x(t, p) ∈ B for each t ≥ τ. See Figure 4.3. The equation λFt (p) + (1 − λ)(p − p∗ ) = 0 is equivalent to p − p∗ = λ(x(t, p) − p∗ ). This cannot hold if t ≥ τ, λ ∈ [0, 1] and p ∈ 𝜕B. In consequence Ft and the translation p 󳨃→ p − p∗ are homotopic. The common degree is clearly 1.

Figure 4.3: The absorbing ball B.

80 | 4 Index of stable fixed points and periodic solutions Third step. Conclusion Consider the continuous map H : cl(B) × [0, τ] → ℝd , − 1 Ft (p) if t > 0, H(t, p) = { t X(p) if t = 0. For t ≠ 0 this map can be expressed as t

1 H(t, p) = − ∫ X(x(s, p))ds. t 0

The continuity of H at t = 0 can be proved using this formula. From the second step we know that H(t, p) ≠ 0 if p ≠ p∗ . Then we can apply again the homotopy property to deduce that 1 deg(X, B) = deg(− Fτ , B). τ The linear map L(x) = − 1t x has determinant

(−1)d . td

Since − τ1 Fτ = L ∘ Fτ we deduce that

1 deg(− Fτ , B) = (−1)d deg(Fτ , B) = (−1)d . τ Let us now go to the general situation and assume that p∗ is a stable and isolated equilibrium, not necessarily attracting nearby solutions. In dimension d = 1 this distinction is irrelevant, since any stable and isolated equilibrium is asymptotically stable. This follows from the description in the line of all possible dynamics around an isolated equilibrium. Let δ > 0 be a number such that X(x) does not vanish if 0 < |x − p∗ | < δ. Let us distinguish four possible cases: (i) (x − p∗ )X(x) < 0,

(ii) (x − p∗ )X(x) > 0,

(iii) X(x) < 0,

(iv) X(x) > 0,

with x ∈ ]p∗ − δ, p∗ + δ[, x ≠ p∗ . The properties of the degree in one dimension imply that the index of X at p∗ is −1 in case (i), +1 in case (ii) and 0 in cases (iii) and (iv). The corresponding phase portraits show that p∗ is stable if and only if the index is −1. See Figure 4.4.

Figure 4.4: Phase portraits on the line.

A similar equivalence, now with index +1, does not hold in dimension d = 2. This can be shown with a simple example. The origin is a repeller for the vector field X :

4.2 The index of a stable equilibrium | 81

ℝ2 → ℝ2 , X(x1 , x2 ) = (x1 , x2 ), in consequence p∗ = 0 is isolated and unstable. The normalization property of degree implies that this index is +1. In the next result we show that there is still some connection between index and stability. Proposition 8. Assume that d = 2 and p∗ is an isolated and stable equilibrium. Then the index of X at p∗ is +1. The proof of this result will require of several preliminary steps. In contrast to the line, in the plane there are many different types of isolated and stable equilibria. Nevertheless, they can be classified in two categories: either they are attractors or they are surrounded by closed orbits. See Figure 4.5.

Figure 4.5: Attractors or closed orbits.

Lemma 13. Assume that d = 2 and p∗ is an isolated and stable equilibrium. Then either p∗ is asymptotically stable or there exists a sequence of closed orbits {Γn }n≥0 with p∗ ∈ Ri (Γn ) and diam(Γn ) → 0 as n → ∞. Exercise 46. Given two vector fields X1 , X2 : G ⊂ ℝ2 → ℝ2 under the conditions of this section, we say that they are orbitally equivalent if there exists a homeomorphism of G mapping orbits of ẋ = X1 (x) onto orbits of ẋ = X2 (x). Construct infinitely many nonequivalent vector fields having a unique equilibrium which is stable. Given p ∈ G with ω(p) = +∞, the ω-limit set of the associated orbit is defined as Lω (p) = ⋂ cl({x(t, p) : t ≥ τ}). τ>0

Note that this set is contained in cl(G). Exercise 47 will be employed in the proof of the above lemma. Exercise 47. Assume that p∗ is a stable equilibrium and p ∈ G with ω(p) = +∞ and p∗ ∈ Lω (p). Then Lω (p) = {p∗ }. Proof of Lemma 13. Consider a sequence of positive numbers {rn } with rn → 0 and let Bn denote the closed ball centered at p∗ of radius rn . For n large enough we know that X(x) ≠ 0 if x ∈ Bn \ {p∗ }. Also, from the stability of p∗ , we can find a neighborhood

82 | 4 Index of stable fixed points and periodic solutions 𝒰n of p∗ such that x(t, p) ∈ Bn if p ∈ 𝒰n and t ≥ 0. In consequence Lω (p) ⊂ Bn for any p ∈ 𝒰n . We distinguish two cases: (i) Lω (p) = {p∗ } for each p ∈ 𝒰n (ii) There exists a point pn ∈ 𝒰n with Lω (pn ) ≠ {p∗ }.

In the first case the whole neighborhood 𝒰n is attracted by p∗ and the equilibrium is asymptotically stable. In the second case we apply Exercise 47 to deduce that p∗ ∈ ̸ Lω (pn ). In consequence Lω (pn ) is a limit set without equilibria. The Poincaré– Bendixson theorem can be applied to deduce that Γn = Lω (pn ) is a closed orbit. This orbit is contained in Bn and surrounds an equilibrium. Therefore diam(Γn ) ≤ 2rn and p∗ ∈ Ri (Γn ). We are ready to prove Proposition 8. If p∗ is asymptotically stable then Proposition 7 is applicable. Otherwise p∗ is surrounded by closed orbits. Let Γn be one of these orbits and assume that the vector field X does not vanish on (Ri (Γn ) ∪ Γn ) \ {p∗ }. This is always the case if the diameter of Γn is small enough. We can invoke Proposition 17 in the appendix on degree theory to deduce that deg(X, Ri (Γn )) = 1. This degree coincides with the index of X at p∗ . In dimension d ≥ 3 the index at an isolated and stable equilibrium can take any integer value. More precisely, given d ≥ 3 and k ∈ ℤ, there exists a C ∞ vector field X : ℝd → ℝd having a unique equilibrium which is stable and such that the index of X at p∗ is k. This result was proved by Bonatti and Villadelprat1 using the theory of plugs due to Wilson. To illustrate this technique we will present a concrete example in the next section. To make the construction as intuitive as possible we will restrict to the case d = 3, k = 1 and the vector field will be only Lipschitz-continuous. Exercise 48 shows that the restriction d = 3 is not very serious. Exercise 48. Assume that X : ℝ3 → ℝ3 is a continuous vector field with uniqueness for the associated initial value problem and having an isolated and stable equilibrium with index k. For each d ≥ 4 construct a vector field Z : ℝd → ℝd having an isolated and stable equilibrium with index (−1)d+1 k.

4.3 The plug construction Consider the vector field X∗ : ℝ3 → ℝ3 ,

X∗ (x1 , x2 , x3 ) = (0, 0, 1).

1 Indeed they proved a stronger result because in their construction p∗ was perpetually stable.

4.3 The plug construction

| 83

Figure 4.6: The parallel flow.

The orbits of the associated flow are straight lines parallel to the x3 -axis. See Figure 4.6. We want to produce local modifications of this vector field leading to orbits that do not escape to infinity. If we are allowed to introduce equilibria, this is an easy task. We can consider a non-negative C ∞ function λ : ℝ3 → ℝ satisfying λ(x) = 1 if ‖x‖ ≥ 2δ and ̂ λ(x) = 0 if ‖x‖ ≤ δ for some δ > 0. The modified vector field X(x) = λ(x)X∗ (x) has many bounded orbits (in the future or in the past). In particular, the solutions with initial conditions lying in x1 (0)2 + x2 (0)2 ≤ δ2 , x3 (0) ≤ 0 will satisfy x1 (t)2 + x2 (t)2 ≤ δ2 , x3 (0) ≤ x3 (t) ≤ 0 if t ≥ 0. The interesting feature of the technique we are going to describe is the absence of equilibria in the modified vector field. Let us fix numbers 0 < r < R and a compact interval [a, b]. Consider the cylindrical region P = {(x1 , x2 , x3 ) ∈ ℝ3 : x12 + x22 ≤ R2 , x3 ∈ [a, b]} and the disk D = {(x1 , x2 , x3 ) ∈ ℝ3 : x12 + x22 ≤ r 2 , x3 = a}. We claim that it is possible to construct a Lipschitz-continuous vector field X̂ ∗ : P → ℝ3 with the following properties: (i) X∗ and X̂ ∗ coincide in a neighborhood of the boundary 𝜕P. (ii) All solutions of the system ẋ = X̂ ∗ (x) with initial condition in D remain in P in the future; that is, x(0) ∈ D implies x(t) ∈ P if t ≥ 0. (iii) X̂ ∗ (x) ≠ 0 for every x ∈ P.

Figure 4.7: A modified flow without equilibria.

The couple (P, X̂ ∗ ) will be called a plug. See Figure 4.7. Later we will explain how to construct X̂ ∗ but now we want to illustrate the use of this device. A first observation is that plugs do not exist in two dimensions.

84 | 4 Index of stable fixed points and periodic solutions Exercise 49. Consider P = {(x1 , x2 ) ∈ ℝ2 : |x1 | ≤ R, x2 ∈ [a, b]} and D = {(x1 , x2 ) ∈ ℝ2 : |x1 | ≤ r, x2 = a}. Prove that there is no Lipschitz-continuous vector field X̂ ∗ : P → ℝ2 satisfying the conditions (i), (ii) and (iii) with X∗ (x1 , x2 ) = (0, 1). The plug can be employed to modify X∗ . Let us define Xnew : ℝ3 → ℝ3 ,

Xnew (x) = {

X̂ ∗ (x) if x ∈ P,

X∗ (x) if x ∈ ℝ3 \ P.

The new vector field is Lipschitz-continuous; it has no equilibria and the orbits passing through D will remain in P for the future. We have inserted a plug in X∗ . Assume now that X : G ⊂ ℝ3 → ℝ3 is a C ∞ vector field which is equivalent to the restriction of X∗ on some open set G∗ ⊂ ℝ3 . This means that there exists a diffeomorphism φ : G → G∗ such that the change of variables u = φ(x) transforms the system ẋ = X(x), x ∈ G in u̇ = X∗ (u), u ∈ G∗ . The plug (P, X̂ ∗ ) can be transported from the u-space to the x-space. Assuming P ⊂ G∗ , we define 𝒫 = φ−1 (P), 𝒟 = φ−1 (D) and X̂ : 𝒫 → ℝ3 ,

̂ X(x) = φ󸀠 (x)−1 X̂ ∗ (φ(x)).

We will say that (𝒫 , X)̂ is a deformed plug, or simply a plug. As an example we consider the linear vector field X(x1 , x2 , x3 ) = (−x1 , −x2 , x3 )

(4.8)

and the map u = φ(x) given by the formulas u1 = x1 x3 ,

u2 = x2 x3 ,

u3 = ln x3 .

Then φ is a diffeomorphism from G = ℝ2 × ]0, ∞[ onto G∗ = ℝ3 and a simple computation shows that it is an equivalence between X|G and X∗ . See Figure 4.8. The set 𝒫 is determined by a surface of revolution and 𝒟 is a disk of radius re−a . See Figure 4.9. From now on the vector field X is given by equation (4.8). The origin is the only equilibrium2 and it has index +1. This equilibrium is unstable and we plan to insert a sequence of deformed plugs making it stable without changing the index. 2 Note that X can be seen as a linear map with positive determinant.

4.3 The plug construction

| 85

Figure 4.8: Equivalence of flows.

Figure 4.9: Transport of a plug.

Let 𝒫1+ and 𝒫1− be symmetric plugs inserted in X. They go around the x3 -axis and lie on the half-spaces {x3 > 0} and {x3 < 0}, respectively. In both plugs the modified vector field is denoted with the same symbol, X̂ 1 : 𝒫1+ ∪ 𝒫1− → ℝ3 . See Figure 4.10. The next step in the construction deals with the cylinder K1 = {(x1 , x2 , x3 ) ∈ ℝ3 : x12 + x22 = r12 , x3 ∈ [−h1 , h1 ]} where r1 is the radius of the disks 𝒟1± and h1 is the height. See Figure 4.11. We insert plugs inside K1 . Define 𝒫2 = ϵ1 𝒫1 , ±

±

where ϵ1 is a positive number such that the inclusion 𝒫2 ∪ 𝒫2 ⊂ K1̂ +

−

Figure 4.10: The first step of the construction.

86 | 4 Index of stable fixed points and periodic solutions

Figure 4.11: The cylinder K1 .

holds. Here K̂ 1 = {(x1 , x2 , x3 ) ∈ ℝ3 : x12 + x22 < r12 , x3 ∈ ]−h1 , h1 [}. Define 1 X̂ 2 (x) = ϵ1 X̂ 1 ( x), ϵ1

x ∈ 𝒫2+ ∪ 𝒫2− .

This is a Lipschitz-continuous vector field with the same Lipschitz constant as X̂ 1 and (𝒫2± , X̂ 2 ) is a plug for X. It is important to observe that X and X̂ 2 coincide in a neighborhood of 𝒫2± because the vector field X is positively homogeneous; that is, X(λx) = λX(x) for each λ > 0. The previous process can be repeated infinitely many times to obtain a sequence ± ± of plugs (𝒫n± , X̂ n ) with 𝒫n+1 = ϵn 𝒫n± , X̂ n+1 (x) = ϵn X̂ n ( ϵ1 x). The inclusion 𝒫n+1 ⊂ K̂ n holds n

and we can assume that 0 < ϵn < 21 . See Figure 4.12. Define X̂ ∞ : ℝ3 → ℝ3 ,

X̂ n (x) if x ∈ 𝒫n+ ∪ 𝒫n− , for some n ≥ 1, X̂ ∞ (x) = { + − X(x) if x ∈ ℝ3 \ ⋃∞ n=1 (𝒫n ∪ 𝒫n ).

It is clear that X̂ ∞ is continuous over ℝ3 \ {0}. To prove the continuity at the origin we observe that, for each n ≥ 2, M ‖X̂ n (x)‖ ≤ ϵn−1 ⋅ ϵn−2 ⋅ ⋅ ⋅ ϵ2 ⋅ ϵ1 M < n−1 , 2

x ∈ 𝒫n+ ∪ 𝒫n− ,

where M := maxx∈𝒫1+ ‖X̂ 1 (x)‖. In consequence, X̂ ∞ (x) → 0 as x → 0. Indeed, X̂ ∞ is

Lipschitz-continuous. To prove this stronger claim we pick two points x, y ∈ ℝ3 , x ≠ y and distinguish three situations. Assume first that the segment [x, y] does not contain the origin. Then this segment will cross a finite number of cells 𝒫n± and ‖X̂ ∞ (x) − X̂ ∞ (y)‖ ≤ L‖x − y‖

4.3 The plug construction

| 87

Figure 4.12: The complete construction.

where L = max{1, L1 } and L1 is a Lipschitz constant for X̂ 1 . Here we are using the fact that L1 is a common Lipschitz constant for all the vector fields X̂ n . Assume now that the origin lies in the interior of the segment [x, y]. Select a point ξ ∈ ℝ3 such that 0 ∈ ̸ [x, ξ ] ∪ [ξ , y] and ‖x − ξ ‖ + ‖ξ − y‖ ≤ 2‖x − y‖. The previous case can be applied to the segments [x, ξ ] and [ξ , y] to obtain ‖X̂ ∞ (x) − X̂ ∞ (y)‖ ≤ ‖X̂ ∞ (x) − X̂ ∞ (ξ )‖ + ‖X̂ ∞ (ξ ) − X̂ ∞ (y)‖ ≤ 2L‖x − y‖. Finally, if x = 0 we take a sequence xn ∈ ℝ3 \ {0} with xn → 0 and apply the previous cases to obtain ‖X̂ ∞ (xn ) − X̂ ∞ (y)‖ ≤ 2L‖xn − y‖. The continuity of X̂ ∞ at the origin allows us to pass to the limit to conclude that 2L is a Lipschitz constant for X̂ ∞ . We know that there is uniqueness for the initial value problem associated to ẋ = X̂ ∞ (x) and, by construction, p∗ = 0 is the only equilibrium. We claim that the origin is a stable equilibrium with index +1. To prove the stability we go back to the first step of the construction and recall the geometry of the vector field X on the cylinder K1 . See Figure 4.13. Since X points towards K̂ 1 on x12 + x22 = r12 , |x3 | < h1 , every orbit of Ẋ = X̂ ∞ (x) exiting from K̂ 1 will pass through one of the disks 𝒟1± . This orbit will remain in 𝒫1± by the properties of the plug. Summing up, every orbit starting in K̂ 1 will remain in the future inside K̂ 1 ∪ 𝒫1+ ∪ 𝒫1− . This property is reproduced at each step, starting with the cylinder Kn . The stability of p∗ = 0 follows.

88 | 4 Index of stable fixed points and periodic solutions

Figure 4.13: The geometry of X on the cylinder K1 .

To compute the index we consider a large open ball B such that ∞

⋃ (𝒫n+ ∪ 𝒫n− ) ⊂ B.

n=1

Then X̂ ∞ = X on the boundary 𝜕B and the standard properties of the degree imply that deg(X̂ ∞ , B) = deg(X, B) = 1. By excision this is the index of X̂ ∞ at p∗ = 0. Exercise 50. Consider the vector field X : ℝ3 → ℝ3 ,

X(x1 , x2 , x3 ) = (−x1 , −x2 , |x3 |).

Draw the associated phase portrait and modify the previous construction to obtain a vector field X̂ ∞ : ℝ3 → ℝ3 which is Lipschitz-continuous, X̂ ∞ (0) = 0, X̂ ∞ (x) ≠ 0 if x ≠ 0 and p∗ = 0 is stable and has index 0.

4.4 Some dynamical insights Following the ideas of Fuller we will make an explicit construction of a plug. The vector field X̂ ∗ : P → ℝ3 will be Lipschitz-continuous. The interested reader can modify it to obtain a C ∞ plug. Consider a disk of radius r > 0 D∗ = {(x1 , x2 ) ∈ ℝ2 : x12 + x22 ≤ r 2 } and a solid cylinder with height h > 0, Π = D∗ × [0, h]. We are going to construct a Lipschitz-continuous vector field Y : Π → ℝ3 , Y = (Y1 , Y2 , Y3 ) with the properties:

4.4 Some dynamical insights | 89

x1 Y1 (x) + x2 Y2 (x) ≤ 0, Y3 (x) > 0 if x = (x1 , x2 , x3 ) ∈ 𝜕D∗ × [0, h] and Y = (0, 0, 1) on D∗ × {0} and D∗ × {h}. 󸀠 (ii ) All solutions of ẋ = Y(x) with initial condition x(0) in D∗ × {0} will remain in Π for t ≥ 0. (iii󸀠 ) Y(x) ≠ 0 if x ∈ Π.

(i󸀠 )

Once this vector field has been constructed it is easy to extend it to a larger cylinder P = {(x1 , x2 , x3 ) ∈ ℝ3 : x12 + x22 ≤ R2 , x3 ∈ [−ϵ, h + ϵ]} to produce a plug. Exercise 51. Define X̂ ∗ on P \ Π assuming that Y : Π → ℝ3 is already known. The definition of Y : Π → ℝ3 is illustrated in Figure 4.14, where the cylinder has been divided in four pieces Πi = D∗ × [hi−1 , hi ],

i = 1, 2, 3, 4

with 0 = h0 < h1 < h2 < h3 < h4 = h.

Figure 4.14: Fuller’s flow.

90 | 4 Index of stable fixed points and periodic solutions Let us be more precise about this figure. We start with the region Π2 . The associated flow funnels the disk D1 = D∗ × {h1 } into a smaller disk Δ inside D2 = D∗ × {h2 }. This can be achieved in the following way. We fix a point p ∈ int(D∗ ) \ {0}, p = (p1 , p2 ) and a number h3/2 with h1 < h3/2 < h2 . Then we consider a vector field in the plane which is transversal to 𝜕D∗ and such that p is a global attractor. For instance, we can consider the linear system ẋ1 = p1 − x1 , ẋ2 = p2 − x2 . We take a small disk Δ∗ ⊂ int(D∗ ) \ {0} centered at p and observe that D∗ is attracted uniformly by Δ∗ . See Figure 4.15. Let τ > 0 be a time large enough so that all the solutions starting from D∗ at time t = 0 have to enter into Δ∗ at time t = τ.

Figure 4.15: The absorbing disk.

Define Y(x1 , x2 , x3 ) = (p1 − x1 , p2 − x2 , ϵx3 ) if x12 + x22 ≤ r 2 and x3 ∈ [h1 , h3/2 ]. Here ϵ is a positive constant satisfying ϵ
0

if x ∈ Π \ R3 ,

x1 Y1 (x) + x2 Y2 (x) = 0

if x ∈ Π3 ∪ Π4 .

(4.9) (4.10)

Let x(t) be a solution of ẋ = Y(x) with x(0) ∈ D0 and let [0, ω[ be the corresponding maximal interval (to the right). From the properties (i󸀠 ) and (4.9), we deduce that this solution will pass through the disk Δ for some t1 ∈ ]0, ω[. Then x3 (t1 ) = h2 , x1 (t1 )2 + x2 (t1 )2 = ρ2 with r2 ≤ ρ ≤ r3 . From (4.10) we deduce that x1 (t)2 + x2 (t)2 = ρ2 for each t ∈ [t1 , ω[ as long as x(τ) remains in Π3 ∪ Π4 for every τ ∈ [t1 , t]. The closed orbit x12 + x22 = ρ2 , x3 = h3 acts as a barrier and therefore x3 (t) < h3 if t ∈ [t1 , ω[. Then ω = +∞ and the solution remains in Π3 if t ≥ t1 . The condition (ii󸀠 ) holds. Once we have completed the construction of Y : Π → ℝ3 , it is possible to understand the dynamics of the system ẋ = Y(x). The only closed orbits are the concentric circles inside the ring R3 . This is a consequence of the condition (4.9). Every orbit passing through the region r22 ≤ x12 + x22 ≤ r32 , h2 ≤ x3 ≤ h3 has to converge to a circle in R3 . In particular, this is the case of any orbit starting at D0 . One way to prove this convergence is to apply LaSalle’s invariance principle via the Lyapunov function V(x) = −x3 . The remaining orbits will exit Π through D4 . This is the dynamics in the plug. Let us go back to the system ẋ = X̂ ∞ (x) constructed in the previous Section. Since X̂ ∞ is Lipschitz-continuous, the associated flow {ϕt }t∈ℝ is globally defined. The origin

92 | 4 Index of stable fixed points and periodic solutions p∗ = 0 is a stable equilibrium of this flow. The proof of Proposition 6 can be adapted to the continuous setting. In consequence p∗ can be surrounded by a family of positively invariant open neighborhoods. Let {𝒰n } be a sequence with the properties listed in Proposition 6. More precisely, cl(𝒰n+1 ) ⊂ 𝒰n , diam(𝒰n ) → 0 and ϕt (𝒰n ) ⊂ 𝒰n

for each t ≥ 0.

In principle there are many possible families {𝒰n }, we would like to find explicitly one of them. To this end we will follow the evolution of each cylinder Kn . First let us consider the set Σ+ = ⋃ ϕt (𝒟̇ + ) n

where

n

t≥0 3

2

2

2

𝒟̇ n = {(x1 , x2 , x3 ) ∈ ℝ : x1 + x2 < rn , x3 = hn }. +

The map (t, x) 󳨃→ ϕt (x) defines a homeomorphism from [0, ∞[ × 𝒟̇ n+ onto Σ+n . The closure of Σ∗n is a continuum which is not arcwise connected, namely cl(Σ+n ) = ⋃ ϕt (𝒟̇ n+ ) ∪ R̃ 3 , t≥0

where R̃ 3 is a compact subset of R3 composed of concentric circles. See Figure 4.16 for a homeomorphic set.

Figure 4.16: The evolution of the upper disk.

The neighborhood 𝒰n is defined as 𝒰n = 𝒱n ∪ Σn ∪ Σn +

−

with 3

2

2

2

𝒱n = {(x1 , x2 , x3 ) ∈ ℝ : x1 + x2 < rn , |x3 | < hn }.

Then 𝒰n is homeomorphic to an open ball but its closure has a complicated topology. See Figure 4.17 for a homeomorphic set.

4.5 The index of a stable fixed point | 93

Figure 4.17: A positively invariant neighborhood.

4.5 The index of a stable fixed point We extend the results of Section 4.2 to a discrete setting. The differential equation (4.3) will be replaced by the difference equation xn+1 = h(xn ),

xn ∈ G ⊂ ℝd ,

(4.11)

where G is an open set and h : G ⊂ ℝd → ℝd is a continuous map. Throughout the section we also assume that the properties below hold, (h1) h is one-to-one; (h2) h is orientation-preserving. In analogy with Section 3.2, this last condition means that deg(h − h(p), 𝒰 ) = 1 for any 𝒰 , open and bounded set with cl(𝒰 ) ⊂ G and p ∈ 𝒰 . Let p∗ ∈ G be a fixed point of h. We say that p∗ is isolated if there exists some neighborhood 𝒱 such that Fix(h) ∩ 𝒱 = {p∗ }, where Fix(h) = {x ∈ G : h(x) = x}. It is possible to choose 𝒱 with some additional properties: open, bounded and such that 𝜕𝒱 ∩ Fix(h) = 0. In this way we can define the fixed point index of h at p∗ as the following degree: I(h, p∗ ) = deg(id − h, 𝒱 ). We are interested in the possible values of this index when p∗ is a stable fixed point. Let us first discuss the special case of an attractor.

94 | 4 Index of stable fixed points and periodic solutions The fixed point p∗ is an attractor if there exists some neighborhood 𝒱 ⊂ G of p∗ such that every orbit of (4.11) starting at 𝒱 is complete and satisfies xn → p∗

as n → +∞.

We say that the orbit {xn } starts at 𝒱 if x0 ∈ 𝒱 . This orbit is complete if it is well defined for every n ≥ 0. Attractors are always isolated as fixed points. A fixed point is asymptotically stable if it is a stable attractor. Theorem 11. The index of an asymptotically stable fixed point is I(h, p∗ ) = 1. Proof. Let us select a closed ball B centered at p∗ and contained in 𝒱 , the set given by the definition of attractor. Then we know that the successive iterates hn (B) are well defined for each n ≥ 0 and we claim that the following limit holds: lim hn (x0 ) = p∗ ,

n→+∞

uniformly in x0 ∈ B. This is proved in the same way as the first step of the proof of Proposition 7. Let us select an integer N ≥ 1 such that hn (B) ⊂ int(B) if n ≥ N. Then Browder’s principle is applicable (see Theorem 20 in the appendix) and we deduce that deg(id − h, int(B)) = 1. Since p∗ is the only fixed point of h lying in B, we conclude that this degree coincides with the index I(h, p∗ ). Remark. The previous proof only requires the continuity of the map h. Many results in continuous dynamics can be seen as corollaries of more general theorems on discrete dynamics. This is the case of Proposition 7, which can be deduced from Theorem 11. To show this we present an auxiliary result connecting the index of a vector field at an equilibrium with the fixed point index of the corresponding flow. Lemma 14. Let x(t, p) denote the solution of ẋ = X(x), x(0) = p, where the vector field X is under the conditions of Section 4.2. Assume that p∗ is an equilibrium and there exist a number τ > 0 and an open and bounded set 𝒢 ⊂ G with p∗ ∈ 𝒢 and such that ω(p) > τ

and

x(t, p) ≠ p

if t ∈ ]0, τ], p ∈ cl(𝒢 ) \ {p∗ }.

Then deg(id − x(τ, ⋅), 𝒢 ) = (−1)d deg(X, 𝒢 ).

(4.12)

4.5 The index of a stable fixed point | 95

Proof. We employ the notations introduced in the proof of Proposition 7 and consider the map Ft (p) = p − x(t, p). The condition (4.12) implies in particular that Ft (p) ≠ 0 if p ∈ 𝜕𝒢 and t ∈ ]0, τ]. In consequence deg(Ft , 𝒢 ) is constant on ]0, τ]. The argument in the third step of the proof of Proposition 7 shows that 1 deg(X, 𝒢 ) = deg(− Fτ , 𝒢 ) = (−1)d deg(Fτ , 𝒢 ). τ Here we have employed well known properties of the degree. See in particular Exercise 79 of the appendix. The condition (4.12) says that the system has no equilibria lying in cl(𝒢 ) \ {p∗ } or closed orbits passing through cl(𝒢 ) with minimal period T ≤ τ. When p∗ is an asymptotically stable equilibrium of ẋ = X(x), all orbits near p∗ are attracted by this point. This implies that the condition (4.12) will hold when 𝒢 is a small neighborhood of p∗ and τ > 0 is arbitrary. Since p∗ can also be interpreted as an asymptotically stable fixed point of h(p) = x(τ, p), we conclude that (−1)d deg(X, 𝒢 ) = deg(id − h, 𝒢 ) and Proposition 7 follows from Theorem 11. Next we consider stable fixed points which are not necessarily attractors. Again the results will be in parallel with the continuous case, but proofs will be more delicate. Let us first assume that the dimension is d = 1 and the domain G is an open interval. In view of (h1) and (h2), the function h is increasing and continuous. We are also assuming that the fixed point p∗ is isolated. In a neighborhood of p∗ we must be in one of the cases described in Figure 4.18.

Figure 4.18: Index 1 and stability are equivalent if d = 1.

In the first case we observe that the fixed point p∗ is stable and has index +1. In the remaining cases p∗ is unstable and the index is either −1 or 0. As can be expected, the equivalence between index I = 1 and stability breaks down in dimension d = 2. Exercise 52. Describe all linear maps h : ℝ2 → ℝ2 , h(x) = Ax with det A > 0, such that the origin is an unstable fixed point with index I(h, 0) = 1.

96 | 4 Index of stable fixed points and periodic solutions The next result can be seen as a discrete counterpart of Proposition 8. Theorem 12. Assume that d = 2 and h satisfies (h1) and (h2). Let p∗ be a fixed point which is isolated and stable. Then I(h, p∗ ) = 1. At first sight, it could seem possible to follow the strategy designed in Lemma 14 to obtain Proposition 8 as a corollary of the theorem. However, this is not possible, as shown by the example below. Example 4. Consider the vector field X : ℝ2 → ℝ2 ,

X(x1 , x2 ) = (x2 , −x11/3 ).

Although the second component is not locally Lipschitz-continuous, it is possible to prove that there is uniqueness for the initial value problem associated to the autonomous system ẋ = X(x). See Exercise 1. The origin is the unique equilibrium and the function 1 3 E(x1 , x2 ) = x22 + x14/3 2 4 is a first integral. From these facts it is easy to deduce that p∗ is an isolated and stable equilibrium. In consequence Proposition 8 is applicable. The dynamics of this system can be described in terms of the first integral, non-trivial orbits are closed and coincide with the energy levels of E, γσ = {(x1 , x2 ) ∈ ℝ2 : E(x1 , x2 ) = σ},

σ > 0.

Given a solution (x1 (t), x2 (t)) of ẋ = X(x), we can obtain the family of solutions (α3 x1 ( αt ), α2 x2 ( αt )) with α > 0. This observation allows us to conclude that there exists a positive constant k such that each orbit γσ has minimal period T(σ) = kσ 1/4 . Let us fix τ > 0 and consider the map h(p) = x(τ, p) produced by the flow. This continuous map is defined on the whole plane and it satisfies the conditions (h1) and (h2). Now p∗ = (0, 0) is a stable fixed point but Theorem 12 is not applicable to the map h. In fact, since p∗ is not isolated as a fixed point, the index I(h, 0) is undefined. To prove this we observe that the fixed points of h are all the points lying in orbits with period τ. In consequence the minimal period must be nτ , n = 1, 2, . . . , and ∞

Fix(h) = {0} ∪ ⋃ γσ(n) n=1

where kσ(n)1/4 = nτ . Since σ(n) → 0 as n → +∞, the origin is an accumulation point in Fix(h).

4.6 Simply connected invariant neighborhoods and proof of Theorem 12

|

97

In the previous example the vector field was Hölder-continuous but not Lipschitzcontinuous. This is essential if we want to produce a family a closed orbits converging to an equilibrium and having periods Tn → 0. For Lipschitz-continuous vector fields, J. A. Yorke found a deep link between the minimal period and the Lipschitz constant: Assume that X : ℝd → ℝd is a vector field satisfying the condition ‖X(x) − X(y)‖ ≤ L‖x − y‖,

x, y ∈ ℝd ,

where L > 0 and ‖ ⋅ ‖ is the Euclidean norm. Let x(t) be a non-constant periodic solution of ẋ = X(x) with period T > 0. Then T ≥ 2π . L Exercise 53. Show that Proposition 8 becomes a corollary of Theorem 12 when the vector field X is Lipschitz-continuous in a neighborhood of p∗ . As could be expected, Theorem 12 cannot be extended to higher dimensions, at least in the class of continuous maps satisfying (h1) and (h2). This is shown in a second example. Example 5. Given d ≥ 3 and any integer N, we know from Sections 4.2 and 4.3 that there exists a Lipschitz-continuous vector field X : ℝd → ℝd such that the origin is a stable and unique equilibrium of ẋ = X(x) and the index of X at the origin is N. In view of Yorke’s result, we know that all closed orbits have minimal period T ≥ 2π , where L L is a Lipschitz constant. In particular, x(t, p) ≠ p

if t ∈ ]0,

2π [ and p ≠ 0. L

Let us fix τ ∈ ]0, 2π [ and consider the map h(p) = x(τ, p). As in the previous example L we know that h is well defined in the whole space ℝd and satisfies the conditions (h1) and (h2). The fixed point p∗ = 0 is stable and isolated. In contrast to the previous example, now we have a well defined fixed point index I(h, 0) = (−1)d N. This is a consequence of Lemma 14 when it is applied to the vector field X in a neighborhood 𝒢 of the origin such that X(x) ≠ 0 if x ∈ cl(𝒢 ) \ {0}.

4.6 Simply connected invariant neighborhoods and proof of Theorem 12 Stable fixed points are surrounded by positively invariant neighborhoods. This fact was established in Proposition 6 under very general conditions. In this section we will prove that in two dimensions these invariant neighborhoods may enjoy some additional properties. More specifically, they will be open and simply connected. Proposition 9. Assume that h : G ⊂ ℝ2 → ℝ2 is a one-to-one and continuous map defined on some open subset G of ℝ2 and let p∗ = h(p∗ ) be a stable fixed point of h.

98 | 4 Index of stable fixed points and periodic solutions Then there exists a sequence {𝒰k }k≥1 of open and simply connected neighborhoods of p∗ satisfying the properties listed in Proposition 6. Moreover, if p∗ is perpetually stable, the neighborhoods 𝒰k can be chosen with the additional invariance property h(𝒰k ) = 𝒰k . To prove this result we need some preliminaries on planar topology. An open and connected subset of the plane will be called a domain. An arbitrary domain Ω ⊂ ℝ2 can have holes but they can be filled in to produce a new domain Ω̂ which is simply connected. See Figure 4.19.

Figure 4.19: A domain with two holes.

This intuitive procedure can be reformulated in precise terms. Lemma 15. Assume that Ω ⊂ ℝ2 is a domain. Then there exists another domain Ω̂ ⊂ ℝ2 with the following properties: (i) Ω ⊂ Ω;̂ (ii) Ω̂ is simply connected; (iii) Ω̂ is minimal in the following sense: given a simply connected domain ω ⊂ ℝ2 with Ω ⊂ ω, then also Ω̂ ⊂ ω. The third property implies that Ω̂ is unique. This result cannot be extended to higher dimensions, as is illustrated with the following example in three dimensions. Let Ω ⊂ ℝ3 be the open solid torus obtained after rotating the disk {(x, 0, z) ∈ ℝ3 : (x −

2

1 3 ) + z2 < } 4 16

around the z axis. Consider the sets (see Figure 4.20) ω+ = Ω ∪ B+ ,

ω− = Ω ∪ B− ,

Figure 4.20: An open solid torus and a simply connected domain.

4.6 Simply connected invariant neighborhoods and proof of Theorem 12

|

99

where B± = {(x, y, z) ∈ ℝ3 : x2 + y2 + z 2 < 1, ±z > 0}. Then ω+ and ω− are simply connected domains in ℝ3 with ω− ∩ ω+ = Ω. Since Ω is not simply connected, the domain Ω̂ cannot exist. Proof of Lemma 15. Let us first recall a classical result on Jordan curves in the plane.3 Assume that Γ1 and Γ2 are Jordan curves in ℝ2 whose interior regions have non-empty intersection Ri (Γ1 ) ∩ Ri (Γ2 ) ≠ 0, then there is another Jordan curve in ℝ2 , denoted by Γ1 ∨ Γ2 , with the properties (see Figure 4.21) Γ1 ∨ Γ2 ⊂ Γ1 ∪ Γ2 ,

Ri (Γ1 ) ∪ Ri (Γ2 ) ⊂ Ri (Γ1 ∨ Γ2 ).

Figure 4.21: Γ1 ∨ Γ2 is strictly contained in Γ1 ∪ Γ2 .

Given the domain Ω, we consider the family 𝒥Ω of all Jordan curves Γ contained in Ω. The set Ω̂ is defined as Ω̂ = ⋃ Ri (Γ). Γ∈𝒥Ω

From this definition we observe that Ω̂ is an open subset of ℝ2 . Moreover, it contains the domain Ω. Indeed, given any point x ∈ Ω we can find a small closed disk B centered at x and such that B ⊂ Ω. The boundary of the disk, 𝜕B is a curve in the family 𝒥Ω and so the interior of the disk is contained in Ω.̂ In particular, x ∈ Ω.̂ Once we know that (i) holds, it is easy to prove that Ω̂ is arcwise connected, since every point lying in Ri (Γ) can be connected to Γ by a path inside Ω.̂ At this point we know that Ω̂ is a domain satisfying the first property. The proof of (ii) is more delicate. We must show that for any Jordan curve γ ⊂ Ω̂ the set Ri (γ) is also contained in Ω.̂ The definition of Ω̂ and the compactness of γ imply the existence of a finite sub-family Γ1 , . . . , Γn ∈ 𝒥Ω such that n

γ ⊂ ⋃ Ri (Γh ). h=1

3 This is related to a result stated in Section 3.7 on the connected components of the intersection of two disks.

100 | 4 Index of stable fixed points and periodic solutions We are going to prove that, whenever n > 1, it is possible to decrease the number of curves in 𝒥Ω employed to cover γ. Let us assume first that Ri (Γr ) ∩ Ri (Γs ) = 0 for each 1 ≤ r < s ≤ n. Since γ is connected, it must be contained in one of those open regions, say γ ⊂ Ri (Γ1 ), and we only need one curve. Assume now that some of the regions Ri (Γh ) are not disjoint, say Ri (Γ1 ) ∩ Ri (Γ2 ) ≠ 0. Then we can consider the family Γ1 ∨ Γ2 , Γ3 , . . . , Γn , having n − 1 curves. This new family also produces a covering of γ because Ri (Γ1 ) ∪ Ri (Γ2 ) ⊂ Ri (Γ1 ∨ Γ2 ). Since Γ1 ∨ Γ2 is contained in Γ1 ∪ Γ2 , we deduce that Γ1 ∨ Γ2 ∈ 𝒥Ω . Summing up all the previous observations, we conclude that it is always possible to find one curve Γ ∈ 𝒥Ω such that γ ⊂ Ri (Γ). Then Ri (γ) ⊂ Ri (Γ) ⊂ Ω̂ and we conclude that (ii) holds. Finally, we observe that (iii) is an automatic consequence of the construction of Ω.̂ Actually, given a simply connected domain ω with Ω ⊂ ω, the inclusions Γ ⊂ Ω ⊂ ω imply that Ri (Γ) ⊂ ω. The operation of filling the holes is preserved by embeddings. Lemma 16. Given h ∈ ℰ (ℝ2 ), ̂ h(Ω)̂ = h(Ω) for every domain Ω ⊂ ℝ2 . Proof. We know from Lemma 4 that h is open and so h(Ω) is also a domain. Moreover, 𝒥h(Ω) = {h(Γ) : Γ ∈ 𝒥Ω }.

̂ and Ω̂ together with Lemma 6 imply that The method of construction of h(Ω) ̂ h(Ω)̂ = ⋃ h(Ri (Γ)) = ⋃ Ri (h(Γ)) = h(Ω). Γ∈𝒥Ω

Γ∈𝒥Ω

Later it will be convenient to work with a local version of the previous lemma. Exercise 54. Assume that D is a disk in the plane and h : D ⊂ ℝ2 → ℝ2 is one-to-one ̂ for every domain Ω with cl(Ω) ⊂ D. and continuous, then h(Ω)̂ = h(Ω) Proof of Proposition 9. We know from Lemma 4 that the map h is open, then Proposition 6 is applicable to the map M = h and the fixed point x∗ = p∗ . Let us fix an open ball B1 of radius r1 , centered at p∗ and so small that cl(B1 ) ⊂ G. We can find an open neighborhood 𝒱1 of p∗ satisfying h(𝒱1 ) ⊂ 𝒱1 ⊂ cl(𝒱1 ) ⊂ B1 . In the proof of Proposition 6 the set 𝒱1 is obtained as the union of all iterates of a smaller neighborhood. If this neighborhood is connected, it is clear that also 𝒱1 will be connected. Since h(𝒱1 ) ⊂ 𝒱1 , ? the previous Exercise 54 can be invoked to deduce that h(𝒱̂1 ) = h( 𝒱1 ) ⊂ 𝒱̂1 . Let us de1 ̂ fine 𝒰1 = 𝒱1 and consider a second open ball B2 of radius r2 ≤ 2 r1 centered at p∗ with cl(B2 ) ⊂ 𝒰1 . We construct 𝒰2 as before. In particular cl(𝒰2 ) ⊂ B2 . This process can be repeated infinitely many times to produce the sequence {𝒰k }k≥1 . We are ready to prove Theorem 12.

4.7 Some corollaries of Theorem 12

| 101

Proof of Theorem 12. Let us distinguish two cases: 1. Asymptotic stability. If p∗ is asymptotically stable we can apply Theorem 11, which is valid in arbitrary dimension. 2. Stability without attraction. The fixed point p∗ is stable but not asymptotically stable. This case is more delicate. Since p∗ is an isolated fixed point, there exists a neighborhood 𝒱 ⊂ G such that Fix(h) ∩ 𝒱 = {p∗ }. Then we can apply Proposition 9 to find another neighborhood 𝒰 ⊂ 𝒱 such that 𝒰 is a simply connected domain and h(𝒰 ) ⊂ 𝒰 . Notice that, in particular, Fix(h)∩ 𝒰 = {p∗ }. It is well known that every simply connected domain of the plane is homeomorphic to the whole plane, let ψ : 𝒰 → ℝ2 denote such a homeomorphism.4 Define h1 : ℝ2 → ℝ2 ,

h1 = ψ ∘ h ∘ ψ−1 .

This new map has the following properties: (i) h1 ∈ ℰ∗ (ℝ2 ); (ii) q∗ = ψ(p∗ ) is a stable fixed point of h1 which is not asymptotically stable; (iii) Fix(h1 ) = {q∗ }; (iv) I(h1 , q∗ ) = I(h, p∗ ). All these properties are more or less automatic with the help of degree theory (see the appendix). To prove (i) we employ the formula for the degree of a composition and to prove (iv) we employ the excision property and the topological character of the fixed point index. We observe that the properties (ii) and (iii) imply that h1 is not free. Otherwise h1 should have trivial dynamics (see Proposition 3) and so all the points close to q∗ should be attracted by the stable fixed point. In view of this conclusion we can apply Theorem 6 to obtain a Jordan curve Γ ⊂ ℝ2 \ {q∗ } such that deg(id − h1 , Ri (Γ)) = 1. Using (iii) we conclude that q∗ is contained in Ri (Γ) and, by excision, I(h1 , q∗ ) = 1. The proof is complete since we know that the indices of p∗ and q∗ coincide.

4.7 Some corollaries of Theorem 12 Many interesting consequences can be derived from the connections between fixed point index and stability properties. We have selected two of them. 4.7.1 Resonance at the roots of the unity Perhaps the most direct application of Theorem 12 is the obtention of instability criteria via degree theory. We present results whose origin is in a paper by Levi-Civita. 4 This fact was already employed in Section 2.3.

102 | 4 Index of stable fixed points and periodic solutions To motivate our discussion, let us start with a general difference equation xn+1 = F(xn ),

(4.13)

where F : 𝒰 ⊂ ℝ2 → ℝ2 is a C 1 map defined on an open set 𝒰 containing the origin and such that F(0) = 0. The linearized equation at the origin is xn+1 = Lxn ,

(4.14)

where L = F 󸀠 (0) is the Jacobian matrix. Sometimes x = 0 is an unstable fixed point of (4.13) and a stable fixed point of (4.14). It can be said that the instability has been produced by the nonlinear terms. In view of Lyapunov’s first method, this phenomenon can only happen when the spectrum of L intersects 𝕊1 . We will consider the special case of two complex conjugate eigenvalues which are roots of the unity. Equivalently, Lk ≠ I,

1≤k 0 such that ‖HN (x)‖ ≥ μ‖x‖d ,

x ∈ ℝ2 .

The best choice for this constant is μ = minx∈𝕊1 ‖HN (x)‖. For δ > 0 small enough, we deduce from the condition imposed to the remainder that ‖RN (x)‖ ≤

μ d ‖x‖ 2

if ‖x‖ ≤ δ.

Then x = 0 is the only fixed point of F N in Bδ = {x ∈ ℝ2 : ‖x‖ ≤ δ}. In this ball, ‖x − F N (x)‖ ≥ ‖HN (x)‖ − ‖RN (x)‖ ≥

μ d ‖x‖ . 2

We claim that deg(id − F N , int(Bδ )) = deg(HN , int(𝔻)). This can be proved with the help of the homotopy HN (x) + λRN (x), λ ∈ [0, 1]. Equation HN (x) + λRN (x) = 0 has no solutions lying on 𝜕Bδ and therefore deg(id − F N , int(Bδ )) = deg(−HN , int(Bδ )). The excision property and Exercise 79 in the appendix imply that this degree coincides with deg(HN , int(𝔻)). We apply Theorem 12 with h = F N and p∗ = 0. Since the fixed point index I(F N , 0) = deg(id − F N , int(Bδ )) is different from 1, we conclude that x = 0 is unstable as a fixed point of F N . The stability properties of a fixed point with respect to a map and its iterates are the same. Therefore x = 0 is also unstable with respect to F. To emphasize the dependence of the transformed polynomial with respect to the linear map L, we employ the notation HN = HN[L] . This polynomial behaves well with respect to linear conjugacy in L. If ℒ = P −1 LP and ℋ(x) = P −1 H(Px), then

104 | 4 Index of stable fixed points and periodic solutions [ℒ] (x) = P −1 HN[L] (Px). Therefore the conditions (4.18) and (4.19) are invariant unℋN

der linear changes of variable. Every linear map satisfying (4.15) is conjugate to a rotation of angle 2π m where m N and N are co-prime. From now on we assume that L is a rotation. For computations it is convenient to employ complex notation. After the identification z = x1 + ix2 , z = x1 − ix2 , the polynomial H is expressed5 as d

H(z, z) = ∑ aj z d−j z j , j=0

with aj ∈ ℂ. Since we are assuming Lz = αz with αN = 1, it is easy to obtain explicit formulas for HN . We discuss two particular cases. Resonance at the third root of unity (d = 2, N = 3) Assume that z1 = F(z) is C 2 and z1 = ωz + a0 z 2 + a1 zz + a2 z 2 + ⋅ ⋅ ⋅ where ω2 + ω + 1 = 0. Then z = 0 is unstable if a2 ≠ 0. From the definition, H2 (z, z) = 3ωa2 z 2 and the condition (4.18) is equivalent to a2 ≠ 0. In this case the degree of H2 is −2 and (4.19) holds. Resonance at the fourth root of unity (d = 3, N = 4) Assume that z1 = F(z) is C 3 and z1 = iz + a0 z 3 + a1 z 2 z + a2 zz 2 + a3 z 3 + ⋅ ⋅ ⋅ Then z = 0 is unstable if |a1 | < |a3 |. By direct computation, H4 (z, z) = −4i(a1 z 2 z + a3 z 3 ). The condition (4.18) holds if and only if |a1 | ≠ |a3 |. For |a1 | > |a3 | the homotopy −4i(a1 z 2 z + λa3 z 3 ), λ ∈ [0, 1], does not vanish on 𝜕𝔻 and it is enough to compute the degree for λ = 0. This map coincides in the boundary of the unit disk with an orientation-preserving linear map and the degree is 1, Proposition 10 cannot be applied in this case. For |a1 | < |a3 | a similar homotopy argument shows that the degree is −3. 4.7.2 Stability and persistence In the framework of Section 2.4 we consider the system ẋ = X(t, x),

x ∈ ℝ2 ,

(4.20)

5 The notation H(z, z) is traditional in this context, although z and z are not independent variables.

4.7 Some corollaries of Theorem 12

| 105

where X : ℝ × ℝ2 → ℝ2 is continuous and T-periodic in t. As usual the uniqueness for the initial value problem is assumed. We plan to connect the definition of persistence given in Section 2.4 with the existence of a stable periodic solution. To this end we need an additional definition. A T-periodic solution φ(t) is isolated (period T) if there exists some δ > 0 such that if x(t) is another solution with 0 < ‖x(0) − φ(0)‖ < δ, then x(t) is not T-periodic. In the language of Poincaré maps this means that φ(0) is isolated in the set Fix(P). Exercise 55. Construct an example of a system (4.20) having a T-periodic solution which is isolated for the period T but it is not isolated for the period 7T. Corollary 2. Assume that the system (4.20) has a T-periodic solution which is stable and isolated (period T). Then this system is persistent. The proof is immediate. We know from Theorem 12 that there is a disk such that the fixed point index of the Poincaré map is 1. Then we can proceed as in the proof of Theorem 5 in Section 3.10. In the above corollary it is essential to assume that the stable periodic solution is isolated. This is illustrated by the following example. Fix T > 0 and consider the autonomous system ẋ1 = x2 , ẋ2 = −x2 , with phase portrait sketched in Figure 4.22.

Figure 4.22: A continuum of equilibria without persistence.

The trivial solution x1 = 0, x2 = 0, is stable and T-periodic but it is not isolated. In this case there is no persistence because the perturbed system ẋ1 = x2 , ẋ2 = −x2 + ϵ has no T-periodic solutions if ϵ ≠ 0. Exercise 56. Construct a system having persistence and a unique stable T-periodic solution which is not isolated. Asymptotic stability implies persistence in any dimension,6 but stability is not always persistent in dimension d ≥ 3. We are going to construct a system in ℝ3 that is not persistent but it has a unique T-periodic solution which is stable. To this end we go back to Section 4.3 and consider the vector field X̂ ∞ : ℝ3 → ℝ3 constructed in Exercise 50. The origin is the only equilibrium of the autonomous system ẋ = X̂ ∞ (x) and it is stable. The vector field was constructed by inserting plugs along the z axis to the 6 This is a consequence of Theorem 11.

106 | 4 Index of stable fixed points and periodic solutions

Figure 4.23: A stable equilibrium with zero index.

system X(x1 , x2 , x3 ) = (−x1 , −x2 , |x3 |). These plugs accumulate at the origin according to the phase portrait drawn in Figure 4.23. There is a family of closed orbits accumulating at the origin and all of them have period 2π. We fix T < 2π. The method of construction of the plug in Section 4.4 allows us to assume that the third coordinate of X̂ ∞ is non-negative everywhere. The perturbed system ẋ = X̂ ∞ (x) + ϵe3 ,

ϵ > 0, e3 = (0, 0, 1),

has no equilibria or closed orbits because ẋ3 > 0 for every solution. In consequence, the system ẋ = X̂ ∞ (x) can be interpreted as a T-periodic system that is not persistent and has the T-periodic solution φ ≡ 0.

4.8 Global asymptotic stability and extinction of two populations A fixed point of a map is called globally asymptotically stable (g. a. s.) if it is stable and it attracts all orbits. The same definition can be adapted to an equilibrium of a continuous flow. Many techniques have been employed to prove global attraction, in most cases using some geometrical information of the dynamical system (Lyapunov functions, contraction principle, . . .). We present a topological technique applicable only in two dimensions. The key assumption will be the existence of an invariant ray connecting the fixed point to infinity. This condition was introduced by Alarcón, Guíñez and Gutiérrez. Theorem 13. Assume that h ∈ ℰ∗ (ℝ2 ) is a map satisfying Fix(h) = {p∗ } and the conditions below (i) p∗ is stable; (ii) there is no escape to infinity; (iii) there exists an invariant ray connecting p∗ to infinity. Then p∗ is g. a. s.

4.8 Global asymptotic stability and extinction of two populations | 107

The condition (ii) means that lim inf ‖hn (p)‖ < ∞ n→+∞

for each p ∈ ℝ2 . A ray connecting p∗ and ∞ is a continuous and one-to-one path ρ : [0, 1[ → ℝ2 such that ρ(0) = p∗ and limt→1− ‖ρ(t)‖ = ∞. The ray is invariant if h(R) = R, with R = ρ([0, 1[). Before proving the theorem we are going to present some examples showing that all the assumptions are needed. Example 6 (h must be orientation-preserving). Let us start with a decreasing homeomorphism f : ℝ → ℝ satisfying f (−x) = −f (x) if x ∈ ℝ,

f (1) = −1,

−1
0 the map h = ϕτ satisfies all the assumptions excepting (ii). This time p∗ is asymptotically stable but not g. a. s. Finally, we observe that an unstable attractor of the type illustrated in Figure 4.26 will produce a map satisfying all assumptions excepting (i).

Figure 4.26: An unstable attractor.

Exercise 57. Prove the following result using Theorem 13 or using Poincaré–Bendixson theory. Assume that X : ℝ2 → ℝ2 is a continuous and bounded vector field with X(0) = 0, X(x) ≠ 0 if x ≠ 0. In addition assume the uniqueness for the initial value problem associated to the system ẋ = X(x) and the three conditions below, (i) x = 0 is stable; (ii) every solution x(t) satisfies lim inf ‖x(t)‖ < ∞; t→+∞

(iii) there exists a solution x(t) such that x(t) → 0 as t → +∞, ‖x(t)‖ → ∞ as t → −∞. Then x = 0 is g. a. s.

4.8 Global asymptotic stability and extinction of two populations | 109

Exercise 58. Construct an example showing that the result in Exercise 57 cannot be extended to ℝ3 . Proof of Theorem 13. First we observe that the points in the ray R are attracted by p∗ . Indeed R is homeomorphic to [0, ∞[ and the dynamics of h restricted to R is trivial. The only fixed point is p∗ . Since this fixed point is stable we conclude that hn (p) → p∗ as n → +∞ if p ∈ R. See Figure 4.27.

Figure 4.27: The dynamics on the ray.

In principle, we do not know that the remaining forward orbits are bounded and we cannot take for granted that limit sets are compact. However, we know from condition (ii) that Lω (p) is non-empty for each p ∈ ℝ2 . Moreover, from the definition of limit set, we know that Lω (p) is closed. In view of the condition (i), we can adapt the solution of Exercise 47 to prove that p∗ is g. a. s. if and only if p∗ ∈ Lω (p)

for each p ∈ ℝ2 .

In the rest of the proof we proceed by contradiction, assuming that this condition fails for some point, say p∗ ∈ ̸ Lω (q) for some q. Notice that q ∈ ̸ R. We claim that Lω (q) ∩ R = 0.

(4.21)

Otherwise some point ξ ∈ R should belong to Lω (q). Then all the forward iterates hn (ξ ) will also belong to Lω (q). Since p∗ is the limit of this sequence, it should also belong to Lω (q). This is against the definition of q. Consider the sphere 𝕊2 = ℝ2 ∪ {∞}. The set R̂ = R ∪ {∞} is an arc in 𝕊2 . All arcs are tame in 𝕊2 , meaning that there exists a homeomorphism of 𝕊2 mapping R̂ onto the segment joining p∗ to ∞. In consequence 𝕊2 \ R̂ = ℝ2 \ R is homeomorphic to ℝ2 . Let ψ : ℝ2 \ R → ℝ2 be a homeomorphism. Define H : ℝ2 → ℝ2 , H = ψ ∘ h ∘ ψ−1 . Then H ∈ ℰ∗ (ℝ2 ) and Fix(H) = 0. Let us take some point ξ ∈ Lω (q). This limit is taken with respect to h, Lω (q) = Lω (q, h). From (4.21) we know that ξ ∈ ℝ2 \ R and we can define η = ψ(ξ ). For some increasing sequence of positive integers σ : ℕ → ℕ, hσ(n) (q) → ξ and H σ(n) (ψ(q)) → η. In consequence, lim inf ‖H n (ψ(q))‖ < ∞ n→+∞

and Corollary 1 can be applied. This is absurd; by construction H has no fixed points.

110 | 4 Index of stable fixed points and periodic solutions Invariant rays appear naturally in population dynamics. We present an application to a continuous model of interaction between two species having seasonal effects. Consider the system u̇ = uf (t, u, v),

v̇ = vg(t, u, v),

(4.22)

where f , g : ℝ × [0, ∞[ × [0, ∞[ → ℝ are continuous functions satisfying f (t + 1, u, v) = f (t, u, v),

g(t + 1, u, v) = g(t, u, v).

In addition we assume that there is uniqueness for the initial value problem. This system is defined in the first quadrant but it can be extended to the whole plane by symmetry, namely f (t, u, v) = f (t, |u|, |v|),

g(t, u, v) = g(t, |u|, |v|).

The Poincaré map commutes with the group of four elements {Σ1 , Σ2 , Σ3 , Σ4 }, where Σ1 = id, Σ2 (u, v) = (−u, v), Σ3 (u, v) = (−u − v), Σ4 (u, v) = (u, −v). That is, Σi ∘ P = P ∘ Σi ,

i = 1, 2, 3, 4.

The coordinate axes are invariant under P in the following sense: P((ℝ × {0}) ∩ 𝒟) ⊂ ℝ × {0},

P(({0} × ℝ) ∩ 𝒟) ⊂ {0} × ℝ.

Let us assume that there exists a number b > 0 such that f (t, u, v) ≤ b,

g(t, u, v) ≤ b if t ∈ ℝ, u ≥ 0, v ≥ 0.

(4.23)

The solution of (4.22) with initial conditions u(0) = u0 , v(0) = v0 satisfies the integral equations t

t

u(t) = u0 exp(∫ f (s, u(s), v(s))ds),

v(t) = v0 exp(∫ g(s, u(s), v(s))ds).

0

(4.24)

0

As a consequence of (4.23) and (4.24), we obtain the estimate u(t) ≤ u0 ebt ,

v(t) ≤ v0 ebt ,

if t ∈ [0, ω[.

A standard argument in continuation of solutions allows us to conclude that ω = +∞. Since 𝒟 = ℝ2 , the map P is in the class ℰ∗ (ℝ2 ). Exercise 59. Prove that 𝒟 = ℝ2 when (4.23) is replaced by the following condition: there exist a number b > 0 and an increasing function ϕ : [0, ∞[ → [0, ∞[ such that f (t, u, v) ≤ b,

g(t, u, v) ≤ ϕ(u) if t ∈ ℝ, u ≥ 0, v ≥ 0.

4.8 Global asymptotic stability and extinction of two populations | 111

The behavior of the first species in the absence of the second (v = 0) is determined by the scalar equation u̇ = uf (t, u, 0).

(4.25)

The Poincaré map of this equation is just the restriction of P to the ray R = [0, ∞[ × {0}. In all cases there is positive invariance, P(R) ⊂ R. There is invariance, P(R) = R, when the following assumption is satisfied: Every solution of (4.25) is defined in ] − ∞, +∞[.

(4.26)

We will apply Theorem 13 to obtain a criterion for the extinction of the two species. Before stating it we introduce some notation. A function ϕ : ℝ → ℝ is in the class 𝒜− 1 if it is continuous, 1-periodic and the average is not positive, ∫0 ϕ(t)dt ≤ 0. Theorem 14. Assume that (4.23) and (4.26) hold. In addition, assume that u ≡ 0, v ≡ 0 is the unique 1-periodic solution of (4.22) and there exist numbers 0 < σ0 < σ∞ and functions φ0 , γ0 , φ∞ , γ∞ ∈ 𝒜− such that, for t ∈ ℝ, u ≥ 0, v ≥ 0, f (t, u, v) ≤ φ0 (t),

f (t, u, v) ≤ φ∞ (t),

g(t, u, v) ≤ γ0 (t) if u + v ≤ σ0 ,

g(t, u, v) ≤ γ∞ (t) if u + v ≥ σ∞ .

(4.27) (4.28)

Then there is total extinction for the system (4.22); that is, lim u(t) = lim v(t) = 0,

t→+∞

t→+∞

for every solution (u(t), v(t)). The conditions (4.27) and (4.28) say that the corresponding intrinsic rates of increase have negative average when the total population is very small or very large. Figure 4.28 is a naive scheme illustrating the assumptions.

Figure 4.28: The conditions (4.27) and (4.28).

112 | 4 Index of stable fixed points and periodic solutions Proof. From the previous discussions we know that P ∈ ℰ∗ (ℝ2 ) and P(R) = R with R = [0, ∞[ × {0}. Also, by assumption, Fix(P) = {(0, 0)}. To apply Theorem 13 with h = P and p∗ = (0, 0), we must check that (i) and (ii) hold. Let us fix δ > 0 with δ < σ0 e−b . From (4.24), assuming u0 ≥ 0, v0 ≥ 0 and u0 + v0 ≤ δ, u(t) + v(t) ≤ (u0 + v0 )ebt ≤ σ0

if t ∈ [0, 1].

The solution is inside the region where (4.27) is valid and we can combine this inequality with the identity (4.24) to deduce that t

u(t) ≤ u0 e∫0 φ0 (s)ds ,

t

v(t) ≤ v0 e∫0 γ0 (s)ds

if t ∈ [0, 1].

In particular u1 ≤ u0 , v1 ≤ v0 if (u1 , v1 ) = P(u0 , v0 ). We have proved that (i) holds because the squares |u| + |v| ≤ r are positively invariant under P if r ≤ δ. To prove (ii) we take any solution in the first quadrant and distinguish two cases: Case 1. There exists τ > 0 such that u(t) + v(t) ≥ σ∞ if t ≥ τ. The condition (4.28) is applicable along the solution if t ≥ τ. We conclude that, for large n, 1

un+1 = u(n + 1) ≤ u(n)e∫0 φ∞ (s)ds ≤ u(n) = un . Similarly, vn+1 ≤ vn . For large n the positive sequences un and vn are monotone nonincreasing and, in particular, they are bounded. We conclude that lim supn→+∞ ‖P n (u0 , v0 )‖ < ∞. Case 2. There exists a sequence tn → +∞ such that u(tn ) + v(tn ) ≤ σ∞ for each n. Let us select an integer σ(n) with tn ≤ σ(n) < tn + 1. Then u(σ(n)) = u(tn )e

σ(n)

∫t

n

f (s,u(s),v(s))ds

≤ σ∞ eb .

A similar estimate holds for v(σ(n)). In this case we conclude that lim inf(un + vn ) ≤ 2σ∞ eb . n→+∞

It must be noticed that there are systems with the property of total extinction and such that the trivial solution is not g. a. s. The phase portrait of Figure 4.29 suggests how to construct unstable attractors in population dynamics.

4.9 Stable fixed points of area-preserving maps In this section we will work with a map h : 𝒰 ⊂ ℝ2 → ℝ2 and a point p∗ ∈ 𝒰 . They satisfy the following properties:

4.9 Stable fixed points of area-preserving maps | 113

Figure 4.29: Extinction without stability.

(h1) 𝒰 is open in ℝ2 , h is continuous, one-to-one and orientation-preserving; (h2) h is area-preserving; that is, λ(h(G)) = λ(G) for each open set G ⊂ 𝒰 , where λ denotes the Lebesgue measure on the plane; (h3) p∗ is a stable7 fixed point of h. Going back to Proposition 6 we know that there is a decreasing sequence {𝒰k }k≥1 of open neighborhoods of p∗ shrinking towards p∗ and positively invariant, h(𝒰k ) ⊂ 𝒰k . At first sight it could be thought that, since h preserves the measure, each of these sets 𝒰k is also negatively invariant. The following example shows that this is not always the case.

Figure 4.30: 𝒰1 is not invariant under h.

We go back to Example 1 in Section 3.9.1 and fix some τ > 0. The map h1 = ψτ can be extended to an open neighborhood 𝒰 of the closed unit disk. Assumptions (h1), (h2) and (h3) are satisfied if we take p∗ = (0, √13 ). Let us pick any point ξ lying in the open segment with end points E± = (±1, 0). The image η = h(ξ ) is also in this segment but closer to E− , ‖η − E− ‖ < ‖ξ − E− ‖. See Figure 4.30. Consider the set 𝒰1 = Δ \ [ξ , E+ ], where Δ is the open unit disk. This set is an open neighborhood of p∗ with h(𝒰1 ) = Δ \ [η, E+ ]. This implies that 𝒰1 is positively invariant but h(𝒰1 ) ≠ 𝒰1 . 7 As usual, stability is understood for the future.

114 | 4 Index of stable fixed points and periodic solutions Exercise 60. Prove that if G ⊂ 𝒰 is open and bounded with h(G) ⊂ G, then G \ h(G) is small in ℝ2 (a set of zero measure and first category). Going back to the previous example, we observe that the open set int(cl(𝒰1 )) = Δ is invariant under h1 . The next result shows that this also occurs in a general situation. Lemma 17. Assume that (h1) and (h2) hold and G is an open and bounded subset of ℝ2 with cl(G) ⊂ 𝒰 and h(G) ⊂ G. Then h(G∗ ) = G∗ , where G∗ = int(cl(G)). Proof. First we notice that Lemmas 3 and 4 are still valid for an embedding defined on 𝒰 . In Lemma 3 the bounded set A must satisfy cl(A) ⊂ 𝒰 . From these Lemmas and the positive invariance of G we deduce that h(G∗ ) = h(int(cl(G))) = int(cl(h(G))) ⊂ int(cl(G)) = G∗ . To prove the other inclusion we first show that G ⊂ cl(h(G)). Indeed the set G \ cl(h(G)) is open and has no intersection with h(G). Therefore λ(G) ≥ λ(G \ cl(h(G))) + λ(h(G)). Since λ(G) = λ(h(G)) we deduce that the open set G \ cl(h(G)) has measure zero. Therefore it is empty. Once we know that the inclusion G ⊂ cl(h(G)) holds, we deduce that cl(G) ⊂ cl(h(G)). The inclusion G∗ ⊂ h(G∗ ) is obtained as before. Incidentally, we notice that in this proof it is not necessary to assume that the map is orientationpreserving. Exercise 61. Assume that Δ is the open unit disk and h : Δ → Δ is continuous, one-toone and conservative. The last property is understood in the following sense: for each non-empty open set W ⊂ Δ there exists an integer n = n(W) ≥ 1 such that hn (W) ∩ W ≠ 0. Let G ⊂ ℝ2 be an open set with cl(G) ⊂ Δ and h(G) ⊂ G. Then h(G∗ ) = G∗ . We can now construct a second family of neighborhoods of p∗ , 𝒱k = int(cl(𝒰k )).

From the previous lemma we know that 𝒱k is invariant under h. Moreover, diam(𝒱k ) = diam(𝒰k ) → 0

as k → ∞

and cl(𝒱k+1 ) = cl(𝒰k+1 ) ⊂ 𝒰k ⊂ 𝒱k . From the existence of this family we deduce an important fact:

4.9 Stable fixed points of area-preserving maps |

115

Stability in the future and perpetual stability are equivalent for fixed points of areapreserving maps. Let us go back to Proposition 9, knowing now that p∗ is perpetually stable. We can find a third family of open neighborhoods {𝒲k }k≥1 which are simply connected and invariant, h(𝒲k ) = 𝒲k . They also satisfy the remaining properties listed in Proposition 6. It is interesting to observe that there are open and bounded subsets 𝒱 of the plane which are simply connected and such that 𝒱∗ = int(cl(𝒱 )) has holes. As an example consider the ring A = {p ∈ ℝ2 : 1 < ‖p‖ < 2} and the ring with a cut 𝒱 = A \ S, where S is the segment connecting (1, 0) and (2, 0). Then 𝒱∗ = A. In general, to construct the sets 𝒲k we first saturate 𝒰k , 𝒱k = int(cl(𝒰k )), and later fill in the holes of 𝒱k , 𝒲k = 𝒱̂k . See Figure 4.31.

Figure 4.31: Construction of 𝒲.

The connections between stability and degree theory were explored in Section 4.5. The isolation of the fixed point was a relevant assumption. Now we are going to prove that, for area-preserving maps, these connections survive even if the fixed point is not isolated. Proposition 11. Assume that (h1), (h2) and (h3) hold. Then one of the following alternatives must hold: (i) There exists a neighborhood 𝒱 of p∗ , 𝒱 ⊂ 𝒰 , such that h is the identity on the set 𝒱 . (ii) There exists a sequence of Jordan curves {Γk }k≥1 satisfying Γk ⊂ 𝒰 ,

Fix(h) ∩ Γk = 0,

Γk → {p∗ },

and deg(id − h, Ri (Γk )) = 1. The convergence of Γk is understood in the sense of Hausdorff. Proof. We assume that (i) does not hold and prove (ii). After taking a sequence {𝒲k }k≥1 as before, we observe that the restriction of h defines a homeomorphism h : 𝒲k → 𝒲k .

116 | 4 Index of stable fixed points and periodic solutions We can assume that h is not the identity on 𝒲k . Since 𝒲k is homeomorphic to the open disk, we can transport the map to Δ and define Hk = ψ−1 k ∘ h ∘ ψk , where ψk : Δ → 𝒲k is a homeomorphism. Then Hk ∈ ℋ∗ (Δ) and it is also possible to pull back the Lebesgue measure on 𝒲k to Δ. Define the Borel measure μk (G) = λ(ψk (G)), for each open set G ⊂ Δ. Then μk is under the conditions of Lemma 11 and Hk is μk -preserving. In particular Hk ≠ id is conservative and Theorem 9 is applicable. For each k there exists a Jordan curve γk ⊂ Δ such that it is enough to define Γk = ψk (γk ) and recall the topological invariance of the fixed point index. Remarks. 1. Using Exercise 61 it is possible to extend Proposition 11 to a class of conservative maps. 2. The assumption (h2) is essential in the previous result. We present an example of a map h and a fixed point p∗ satisfying (h1) and (h3) and such that the degree appearing in (ii) can only take the value 0. Consider the system ẋ1 = x22 ,

ẋ2 = −x1 x2 .

The function x12 + x22 is a first integral and it is easy to prove that the associated flow {ϕt }t∈ℝ is globally defined. The points lying on the horizontal line ℝ × {0} are equilibria and the remaining orbits are heteroclinic connections between (±λ, 0) with λ > 0. See Figure 4.32.

Figure 4.32: The fixed point index is zero.

Let us now fix τ > 0 and define h = ϕτ , p∗ = (0, 0). The assumptions (h1) and (h3) hold. Indeed h ∈ ℋ∗ (ℝ2 ) and p∗ is perpetually stable because the circles centered at the origin are invariant. The set of fixed points of this map is Fix(h) = ℝ × {0}

4.9 Stable fixed points of area-preserving maps |

117

and any Jordan curve Γ ⊂ ℝ2 \ Fix(h) must lie in one of the half planes {x2 > 0} or {x2 < 0}. Then the region Ri (Γ) does not contain fixed points and deg(id − h, Ri (Γ)) = 0. 3.

The interest of Proposition 11 concerns the case of non-isolated fixed points, for otherwise Theorem 12 can be applied. An example of non-isolated stable fixed point is the origin z∗ = 0 for the map (in complex notation) 2

h(z) = eiϕ(|z| ) z, with ϕ(ξ ) = e

− ξ1

z ∈ ℂ,

sin( ξ1 ). The conditions (h1), (h2) and (h3) are satisfied. Indeed,

a direct computation shows that the Jacobian matrix satisfies det h󸀠 (z) = 1 everywhere. Then h is a C ∞ diffeomorphism-preserving area and orientation. The 2 inverse is h−1 (z) = e−iϕ(|z| ) z. The identity |h(z)| = |z| implies that z∗ = 0 is stable. This fixed point is not isolated because the circumferences of radius rn = (nπ)−1/2 , n = 1, 2, . . . , are contained in Fix(h).

The map h in the previous example belongs to C ∞ but it is not analytic. The next result shows that it is not possible to construct similar examples in the class C ω . Theorem 15. Assume that h : G ⊂ ℝ2 → ℝ2 is a real analytic map defined on an open and connected subset G satisfying det h󸀠 (p) = 1 for each p ∈ G.

(4.29)

Let p∗ ∈ G be a stable fixed point, h(p∗ ) = p∗ . Then either h = id on G or p∗ is an isolated fixed point of h. Proof. We proceed by contradiction and assume that h is not the identity on G and p∗ is not isolated in Fix(h). With the notations p = (x, y), h = (h1 , h2 ), we observe that Fix(h) is an analytic set described as the set of zeros φ−1 (0) of the real analytic function 2

2

φ(x, y) = (h1 (x, y) − x) + (h2 (x, y) − y) . From the theory on the local structure of analytic sets in the plane we know that there exists an arbitrarily small disk D with p∗ ∈ int(D) such that D ∩ Fix(h) is a star centered at p∗ . This means that there is a homeomorphism between the pairs (Sn , 0) and (D ∩ Fix(h), p∗ ), where Sn = {(r cos θ, r sin θ) : r ∈ [0, 1], θ = See Figure 4.33.

2πk , 1 ≤ k ≤ n}. n

118 | 4 Index of stable fixed points and periodic solutions

Figure 4.33: The local structure of an analytic set.

In view of the condition (4.29), the inverse function theorem can be applied at the point p∗ and it is not restrictive to assume that h is one-to-one in D. Then all the assumptions of Proposition 11 are satisfied on 𝒰 = int(D). Therefore it is possible to find a Jordan curve Γ ⊂ int(D) with Γ ∩ Fix(h) = 0 and deg(id − h, Ri (Γ)) = 1. This Jordan curve must be contained in a connected component C of D \ Fix(h). The same component will also contain the disk Γ ∪ Ri (Γ); that is, Ri (Γ) ⊂ C ⊂ D \ Fix(h). This is not compatible with the value of the degree of id − h on Ri (Γ). See Figure 4.34.

Figure 4.34: The region inside Γ has no fixed points.

The previous result does not extend to higher dimensions. A rotation in ℝ3 is real analytic and preserves volume and orientation. All points lying on the axis of rotation are fixed, stable and non-isolated.

4.10 Some applications to Hamiltonian systems The flow of a Hamiltonian system is composed of a family {ϕt }t∈ℝ of measure-preserving maps. This well-known theorem, valid in any dimension, is due to Liouville. As a consequence, many of the Poincaré sections associated to a system in two degrees of freedom are area-preserving. The results of Section 4.9 are applicable. We present some corollaries for systems with one degree of freedom and periodic time

4.10 Some applications to Hamiltonian systems | 119

dependence (1.5 degrees of freedom) and also for autonomous systems (2 degrees of freedom).

4.10.1 The number of stable periodic solutions Consider the non-autonomous Hamiltonian system with one degree of freedom q̇ =

𝜕H (t, q, p), 𝜕p

ṗ = −

𝜕H (t, q, p), 𝜕q

(q, p) ∈ ℝ2 ,

(4.30)

where the function H : ℝ × ℝ2 → ℝ, H = H(t, q, p), is real analytic and periodic in t, say H(t + T, q, p) = H(t, q, p), where T > 0 is a fixed number. It is said that the system has an a priori bound if there exists a number B > 0 such that |q(t)| + |p(t)| ≤ B,

t ∈ ℝ,

for each T-periodic solution (q(t), p(t)). A priori bounds appear usually in the proofs of existence of periodic solutions. The following result goes in a different direction. Theorem 16. Assume that the system (4.30) is analytic, T-periodic in t and has an a priori bound. In addition, assume that all solutions starting at time t0 = 0 are defined in [0, ∞[. In other words, 𝒟 = ℝ2 . Then the number of stable T-periodic solutions is finite. The a priori bound is essential. The Hamiltonian function H(t, q, p) = 21 (q2 + p2 ) is analytic and T-periodic for any period. For T = 2π we observe that all solutions are stable and 2π-periodic. The conclusion of the theorem is concerned with stable solutions. There are systems under the conditions of the theorem having a continuum of unstable T-periodic solutions. We present an example where all solutions can be explicitly computed. Assume that T = 2π and H(t, q, p) = (1 + cos t)ϕ(

q 2 + p2 ), 2

where ϕ : ℝ → ℝ is a non-constant real analytic function. In complex notation, z = q + ip, the solutions are 2 󸀠 |z0 | 2

z(t) = e−i(t+sin t)ϕ (

)

z0 ,

z0 ∈ ℂ,

120 | 4 Index of stable fixed points and periodic solutions and the Poincaré map is 2 󸀠 |z0 | 2

P(z0 ) = e−i2πϕ (

)

z0 .

This Hamiltonian system has an a priori bound if and only if there exists some r∗ > 0 such that ϕ󸀠 (r) ∈ ̸ ℤ if r ≥ r∗ . Under this condition, the only stable 2π-periodic solution is q = p = 0. If ϕ󸀠 (r1 ) ∈ ℤ for some r1 ∈ ]0, r∗ [, then all the solutions with initial condition in the circle |z0 |2 = 2r1 are 2π-periodic and unstable. The dynamics of P around this circle is described in Figure 4.35 when ϕ󸀠 (r1 ) = 0, ϕ󸀠󸀠 (r1 ) > 0.

Figure 4.35: Dynamics of the Poincaré map around a circle of fixed points.

To prepare for the proof of the theorem we need a preliminary result on the fixed points of a real analytic map. Lemma 18. Assume that f : ℝ2 → ℝ2 is a real analytic map and the set Fix(P) is bounded. Then there is a finite number of isolated fixed points. Proof. The set of fixed points can be split as Fix(P) = ℐ ∪ 𝒞 , where ℐ is the set of isolated fixed points and 𝒞 = Fix(P) \ ℐ . In the proof of Theorem 15 we discussed the local structure of Fix(P) in the neighborhood of a point of 𝒞 . In particular we can deduce that 𝒞 is open in Fix(P). Therefore ℐ is closed in Fix(P), a compact set. We conclude that ℐ is compact and discrete or, equivalently, finite. Exercise 62. Prove that the previous result cannot be extended to C ∞ maps. We are ready to prove the main result. Proof of Theorem 16. The vector field associated to (4.30) is analytic and therefore the solution depends analytically with respect to initial conditions. In particular, the Poincaré map P : ℝ2 → ℝ2 is analytic. The existence of an a priori bound just says that

4.10 Some applications to Hamiltonian systems | 121

the set Fix(P) is bounded. The previous lemma is applicable and we conclude that the set of isolated fixed points ℐ is finite. To finish the proof we will apply Theorem 15 to show that the set of stable fixed points is contained in ℐ . The condition (4.29) is satisfied on the whole plane as a consequence of the Liouville theorem. Then every stable fixed point of P must be either isolated or isochronous, meaning that P = id in a neighborhood. The second possibility is excluded because the identity principle for analytic functions would imply that P = id everywhere. This is impossible since Fix(P) is bounded. Summing up, we know that every stable fixed point is inside ℐ , a finite set. Remarks on the proof. 1. We know from Section 4.9 that the stability in the future and the perpetual stability are equivalent for area-preserving maps. Hence, they are also equivalent for periodic solutions of (4.30). 2. For simplicity we have assumed that the Hamiltonian function was analytic in the three variables but this is not really needed, the proof works as long as P is analytic. In some applications H is analytic in (q, p) but not in t. The following assumption is sufficient to guarantee that P is real analytic. There exists an open set Ω ⊂ ℂ2 containing ℝ2 such that the function H admits an extension H :ℝ×Ω→ℂ with the following properties: – H is continuous. – For each t ∈ ℝ, the function H(t, ⋅) is holomorphic in Ω. It must be noticed that this assumption implies that the vector field in (4.30) is smooth in the following sense: every partial derivative of H with respect to q and p is continuous in the three variables. In other words, for the integers r, s ≥ 0 the function (t, q, p) ∈ ℝ × Ω 󳨃→

𝜕r+s H (t, q, p) ∈ ℂ 𝜕qr 𝜕ps

is continuous. This is a consequence of Cauchy’s formula, H(t, ξ , η) r!s! 𝜕r+s H (t, q, p) = − 2 ∫ ∫ dξdη, r s 𝜕q 𝜕p (ξ − q)r+1 (η − p)s+1 4π C1 C2

where C1 and C2 are positively oriented Jordan curves around q and p, respectively. The complex version of the theorem on differentiability with respect to initial conditions is applicable. In particular, P is holomorphic in some neighborhood of ℝ2 . In the next examples we will employ the stronger version of Theorem 16 including this alternative assumption.

122 | 4 Index of stable fixed points and periodic solutions Example 9. Consider the forced oscillator ẍ + ω2 x + ϕ(x) = f (t), where ω > 0, ϕ : ℝ → ℝ is bounded and analytic, f : ℝ → ℝ is continuous and 2π-periodic. We claim that the number of stable 2π-periodic solutions is finite if ω ∈ ̸ ℤ. The equation is equivalent to the system (4.30) with q = x, p = ẋ and 1 ω2 2 H(t, q, p) = p2 + q + Φ(q) − f (t)q 2 2 where Φ is a primitive of ϕ. Since ϕ is bounded, all solutions are defined in ]−∞, +∞[. In particular 𝒟 = ℝ2 . To apply Theorem 16 we must find an a priori bound for 2π-periodic solutions. To this end we recall some elementary facts on the linear equation ẍ + ω2 x = g(t), where ω ∈ ̸ ℤ and g : ℝ → ℝ is continuous and 2π-periodic. This equation has a unique 2π-periodic solution. Moreover, this solution has the bound ‖x‖∞ + ‖x‖̇ ∞ ≤ C‖g‖∞ , where C only depends upon ω. Let us go back to the nonlinear equation and assume that x(t) is a 2π-periodic solution. We split it as x(t) = x1 (t) + x2 (t) where xi (t) is the unique 2π-periodic solution of ẍ + ω2 x = gi (t) with g1 = f and g2 (t) = −ϕ(x(t)). Then ‖x‖∞ + ‖x‖̇ ∞ ≤ C[‖f ‖∞ + ‖ϕ‖∞ ], and we have found an a priori bound. 2π Notice that if ω = n ∈ ℤ, n ≥ 1, ϕ ≡ 0 and ∫0 f (t)e−int dt = 0, then all the solutions are 2π periodic and stable. The condition ω ∈ ̸ ℤ is essential in the previous discussion. Example 10. Consider the prey–predator system u̇ = u(a(t) − b(t)v),

v̇ = v(−c(t) + d(t)u),

u > 0, v > 0,

where a, b, c, d : ℝ → ℝ are continuous and 1-periodic. In addition we assume that there exists δ > 0 such that b(t) ≥ δ,

d(t) ≥ δ,

for each t ∈ ℝ.

We will prove that Theorem 16 is also applicable to this system. Using Exercise 59 we can prove that all the solutions starting at t0 = 0 are defined on [0, ∞[. The system can be transformed in a system of the type (4.30) with u = eq , v = ep and H(t, q, p) = a(t)p + c(t)q − b(t)ep − d(t)eq .

4.10 Some applications to Hamiltonian systems | 123

Since we already know that 𝒟 = ℝ2 , it is enough to prove the existence of an a priori bound for the system in canonical coordinates, q̇ = a(t) − b(t)ep ,

ṗ = −c(t) + d(t)eq .

Given a 1-periodic solution (q(t), p(t)), we integrate both equations over a period to obtain 1

1

1

1

∫ a = ∫ be ,

∫ c = ∫ deq .

0

0

0

p

0

(4.31)

From the first identity in (4.31) we deduce the existence of an instant t0 ∈ [0, 1] such that 1

δep(t0 ) ≤ b(t0 )ep(t0 ) = ∫ a ≤ ‖b‖∞ ep(t0 ) . 0

From these inequalities it is easy to obtain an estimate |p(t0 )| ≤ B, where B only depends upon a, b and δ. The second equation and the second identity in (4.31) imply, 1

1

1

1

0

0

0

0

∫ |p|̇ ≤ ∫ |c| + ∫ deq ≤ 2 ∫ |c|. For each t ∈ [0, 1], 1

1

|p(t)| ≤ |p(t0 )| + ∫ |p|̇ ≤ B + 2 ∫ |c|. 0

0

A similar estimate can be obtained for |q(t)|. Similar results can be obtained for many other systems. Exercise 63. Prove that Theorem 16 can be applied to ẍ +

x 1+x 2

= λ sin t, where λ ∈ ℝ.

4.10.2 Stable closed orbits with two degrees of freedom In the space ℝ2 × ℝ2 consider the canonical coordinates (q, p) with q, p ∈ ℝ2 and the Hamiltonian system q̇ =

𝜕H (q, p), 𝜕p

ṗ = −

𝜕H (q, p), 𝜕q

where H : Ω → ℝ is a C 2 -function defined on an open subset Ω of ℝ2 × ℝ2 .

(4.32)

124 | 4 Index of stable fixed points and periodic solutions A closed orbit γ∗ ⊂ Ω is orbitally stable if each neighborhood U ⊂ ℝ2 ×ℝ2 of γ∗ contains another neighborhood V such that every orbit γ passing through V is contained in U; that is, γ ∩ V ≠ 0 implies γ ⊂ U. Lyapunov stability is too restrictive for autonomous systems and orbital stability is more appropriate in this context. This is shown by the paradigmatic example 1 1 H(q, p) = ‖p‖2 − , 2 ‖q‖

Ω = (ℝ2 \ {0}) × ℝ2 ,

corresponding to the Kepler problem. Circular motions are described by the periodic solutions q(t) = reiωt ,

p(t) = iωreiωt

with r 3 ω2 = 1 and ℝ2 ≡ ℂ. The period of these solutions changes with the radius, T(r) = 2πr 3/2 . Exercise 44 can be applied to deduce that these solutions are unstable in the Lyapunov sense. Most people would say that the motion of the Earth around the Sun is stable and experts in celestial mechanics will be able to prove that the orbits γr = {(z, iωz) : z ∈ ℂ, |z| = r}, r > 0, are orbitally stable. In the previous example closed orbits have appeared as members of a family. We will prove something similar for general Hamiltonian systems: orbitally stable closed orbits are never alone. To be precise we first define the notion of a family. A family of closed orbits will be a map λ ∈ I 󳨃→ γλ ⊂ Ω such that I ≠ 0 is an open interval, each γλ is a closed orbit of (4.32) and γλ ≠ γμ if λ ≠ μ. Theorem 17. Assume that γ∗ is an orbitally stable closed orbit of system (4.32). Then, given any neighborhood U of γ∗ , there exists a family of closed orbits {γλ }λ∈I such that γλ ⊂ U for each λ. To prepare for the proof we recall some well-known facts in Hamiltonian dynamics. For simplicity we only discuss our concrete situation and avoid the use of general symplectic manifolds. Points in the phase space Ω ⊂ ℝ2 × ℝ2 will be denoted by z = (q, p). The solution passing through z at time t0 = 0 will be denoted by ϕ(t, z). Given a closed orbit γ∗ and a point z∗ = (q∗ , p∗ ) in γ∗ , we observe that ϕ(T∗ , z∗ ) = z∗ , where T∗ > 0 is the minimal period of ϕ(⋅, z∗ ). The point z∗ is not an equilibrium of (4.32) and therefore some partial derivative of H does not vanish at this point. From now on we assume that 𝜕H (q , p ) ≠ 0. 𝜕q2 ∗ ∗

(4.33)

4.10 Some applications to Hamiltonian systems | 125

The other cases are treated similarly. Define λ∗ = H(q∗ , p∗ ). The equations p2 = (p∗ )2 ,

H(q1 , q2 , p1 , p2 ) = λ∗

define a surface Σ of equations p2 = (p∗ )2 , q2 = ψ(q1 , p1 ), where ψ is a C 2 function defined on an open neighborhood 𝒱 ⊂ ℝ2 of ((q∗ )1 , (p∗ )1 ). After reducing the size of 𝒱 , we can assume that Σ satisfies the additional property Σ ∩ γ∗ = {z∗ }.

(4.34)

Let us consider the implicit function problem ϕ4 (T, z) = 0,

T(z∗ ) = T∗ ,

where ϕ = (ϕ1 , ϕ2 , ϕ3 , ϕ4 ). From the transversality condition (4.33) and the second 𝜕ϕ 𝜕H equation in (4.32), we observe that 𝜕t4 (T∗ , z∗ ) = − 𝜕q (z∗ ) ≠ 0. Let T = T(z) be the 2 solution of this implicit problem, defined on a small neighborhood V ⊂ Ω of z∗ We know that T ∈ C 1 (V) and the definitions of Σ and T(z) imply that ϕ(T(z), z) ∈ Σ if z ∈ Σ ∩ V. Since T∗ is the minimal period of γ∗ , the conditions (4.34) and (4.33) imply that T(z) is the first positive instant where ϕ(t, z) meets Σ. This assertion will hold for points in a small neighborhood of z∗ . Define ℱ : Σ ∩ V → Σ, ℱ (z) = ϕ(T(z), z). See Figure 4.36.

Figure 4.36: The Poincaré section.

It is easy to prove that z∗ is a Lyapunov stable fixed point of ℱ when γ∗ is orbitally stable.8 It is well known that the map ℱ preserves the symplectic form. This means that ω((dℱ )(z)v, (dℱ )(z)w) = ω(v, w),

(4.35)

for each z ∈ Σ ∩ V and v, w ∈ Tz (Σ). The symplectic form ω is defined by ω(v, w) = ⟨v, Jw⟩,

v, w ∈ ℝ4 ,

8 The converse is false. Lyapunov stability of the fixed point only implies isoenergetic orbital stability for the closed orbit.

126 | 4 Index of stable fixed points and periodic solutions 0 −I2 0 ).

where J = ( I2 v1 = s1 ,

The tangent space Tz (Σ) is described by the parametric equations

v2 =

𝜕ψ 𝜕ψ (q , p )s + (q , p )s , 𝜕q1 1 1 1 𝜕p1 1 1 2

v3 = s2 ,

v4 = 0,

(s1 , s2 ∈ ℝ).

It is easily seen that the symplectic form on this subspace is given by the formula ω(v, w) = − det(v|̂ w)̂ if v, w ∈ Tz (Σ) and v̂ = (v1 , v3 ), ŵ = (w1 , w3 ).

(4.36)

After these preliminaries we prove the main result. Proof of Theorem 17. Consider the projection Π : Σ → 𝒱 ⊂ ℝ2 , Π(z) = (q1 , p1 ). It is a diffeomorphism whose inverse is Π−1 : 𝒱 → Σ, Π−1 (q1 , p1 ) = (q1 , ψ(q1 , p1 ), p1 , (p∗ )2 ). We define h : 𝒲 ⊂ ℝ2 → ℝ2 ,

h = Π ∘ ℱ ∘ Π−1 ,

where 𝒲 is an open neighborhood of ((q∗ )1 , (p∗ )1 ) contained in 𝒱 and also in the projection of V to the plane (q1 , p1 ). From (4.35) and (4.36) we deduce that h satisfies det h󸀠 = 1

in 𝒲 .

The inverse function theorem allows us to find a small neighborhood 𝒰 such that h : 𝒰 ⊂ ℝ2 → ℝ2 satisfies properties (h1), (h2) and (h3) of Section 4.9, with the stable fixed point ((q∗ )1 , (p∗ )1 ). We can apply Proposition 11 to find two possible situations: (i) h = id in a neighborhood of ((q∗ )1 , (p∗ )1 ); (ii) there exists a Jordan curve Γ ⊂ 𝒰 arbitrarily close to ((q∗ )1 , (p∗ )1 ) such that deg(id− h, Ri (Γ)) = 1. The procedure to construct the family of closed orbits is different in each case. Assume first that (i) holds and consider the arc in Σ, zλ = ((q∗ )1 + λ, ψ((q∗ )1 + λ, (p∗ )1 ), (p∗ )1 , (p∗ )2 ) for λ close to 0. We know that h((q∗ )1 + λ, (p∗ )1 ) = ((q∗ )1 + λ, (p∗ )1 ) or, equivalently,

ℱ (zλ ) = zλ . The solution ϕ(t, zλ ) is periodic with period T(zλ ) and

γλ = {ϕ(t, zλ ) : t ∈ [0, T(zλ )]} is a closed orbit. After restricting the size of λ we can assume that γλ is contained in U because limλ→λ∗ γλ = γ∗ in the Hausdorff metric. Since T(zλ ) is the first positive time when γλ and Σ meet, we conclude that γλ ∩ Σ = {zλ }. In consequence γλ ≠ γμ if λ ≠ μ and {γλ } is a family of closed orbits. Let us now assume that (ii) holds. For λ close to λ∗ , the equations p2 = (p∗ )2 ,

H(q, p) = λ

4.11 Bibliographical remarks | 127

define a surface Σλ with equations p2 = (p∗ )2 , q2 = ψλ (q1 , p1 ) where ψ = ψ(λ, q1 , p1 ) is C 2 on [λ∗ − ϵ, λ∗ + ϵ] × 𝒱 . We define ℱλ : Σλ ∩ V → Σλ ,

ℱλ (z) = ϕ(T(z), z).

The fixed points of ℱλ lead to closed orbits at the energy level H = λ. Again the projection Πλ : Σλ → 𝒱 is a diffeomorphism and we can define the family of maps in the plane hλ : 𝒱 ⊂ ℝ2 → ℝ2 ,

hλ = Πλ ∘ ℱλ ∘ Π−1 λ .

We select the Jordan curve Γ close enough to ((q∗ )1 , (p∗ )1 ) and the parameter λ near λ∗ so that ϕ(t, z) ∈ U if t ∈ [0, T(z)], z = Π−1 λ (q1 , p1 ) with (q1 , p1 ) ∈ Γ ∪ Ri (Γ) and λ ∈ [λ∗ − δ, λ∗ + δ]. The stability of the degree under small perturbations implies that, for |λ − λ∗ | small, Fix(hλ ) ∩ Γ = 0 and deg(id − hλ , Ri (Γ)) = 1. In consequence hλ has a fixed point lying in Ri (Γ). The corresponding fixed point of ℱλ will produce a closed orbit γλ ⊂ U. Moreover, the map λ 󳨃→ γλ is one-to-one because each γλ is contained in a different energy level H = λ. Exercise 64. Prove that the Hamiltonian system associated to the function 1 1 2 H(q1 , q2 , p1 , p2 ) = q12 p1 + p31 + (p31 + p1 )(q22 + p22 − 1) + (q22 + p22 ) 3 2 has a unique closed orbit.

4.11 Bibliographical remarks As regards Section 4.1, different concepts in stability theory can be found in the books [32, 13, 68]. The characterization of perpetual stability in terms of invariant neighborhoods can be found in Section 25 of the book [121]. As far as I know the origin of the proof of Proposition 6 is in Section 42 of the famous paper [14]. Birkhoff suggests to compare his method of proof with some previous work by Poincaré concerning the recurrence theorem. More precisely, Birkhoff refers to pages 149–151 of [107]. Both proofs deal with the successive iterates of an open set. See also Sections 132 and 133 of the book by Wintner [134]. As regards Section 4.2, the books by Krasnosel’skiĭ [63] and by Krasnosel’skiĭ and Zabreĭko [64] contain sections devoted to the index of a stable solution. They have been a relevant motivation for the authors who have considered this question later. The result on the index of an asymptotically stable equilibrium (Proposition 7) can be found in [64], Theorem 52.1 and in [125]. The result on stable equilibria in two dimensions (Proposition 8) can be found in [125]. It is also stated in Section 52.4 of [64] but the proof is left to the reader.

128 | 4 Index of stable fixed points and periodic solutions As regards Section 4.3, the notion of a plug (1-annihilator) was introduced by Wilson in [132]. In that paper, plugs were employed to construct examples of vector fields without equilibria and such that all limit sets were closed orbits or invariant tori of appropriate dimension. The phase space was any compact manifold without boundary having zero Euler characteristic. In Wilson’s original formulation, plugs were defined in the C ∞ class. For simplicity we have worked with the less smooth class of Lipschitz-continuous vector fields. Plugs with additional properties have been constructed by several authors. The reader can find more information in the survey paper [66]. The existence of stable equilibria with index different from (−1)d was discovered by Bobylev and Krasnosel’skiĭ in [15]. See also the book [64] and the paper [43]. The idea in [15] was to construct a vector field in ℝd with a unique equilibrium p∗ and a family of positively invariant neighborhoods {Mn } shrinking towards p∗ . The sets Mn are homeomorphic compact and connected manifolds of dimension d. The boundary is a compact and connected submanifold of dimension d − 1. The Poincaré–Hopf theorem for vector fields (see [50]) implies that the index is (−1)d χ(M0 ), where χ(M0 ) is the Euler characteristic of M0 . The well-known classification of compact surfaces imposes the constraint χ(M0 ) ≤ 1 in dimension d = 3. For this reason the examples in [15] in dimension 3 have index at least −1. The construction with plugs by Bonatti and Villadelprat [16] removed this restriction, proving that all the indices are also possible in dimension d = 3. The construction we have presented can be considered as a particular case of the construction in [16]. See also [133]. For simplicity the stability of the equilibrium is only for the future and the vector field is Lipschitz-continuous. The original construction in [16] deals with perpetual stability and C ∞ vector fields. For real analytic vector fields it is not known if the index of a stable equilibrium can take any value different from (−1)d . As regards Section 4.4, Figure 4.14 is an adaptation of the drawing by Fuller in [48]. This paper has only two pages but it contains many new ideas. In the first part Fuller constructed a flow on a solid torus with the following properties: there are no equilibria and all closed orbits are contractible. The reader can visualize this flow by gluing the top and bottom boundaries of the cylinder of Figure 4.14. In the second part of [48] there is a construction of a flow on a 4-manifold (product of a 3-cell and 𝕊1 ) without equilibria or closed orbits. This second construction was extended by Wilson in [132] to general manifolds with zero Euler characteristics. In the process he introduced the notion of plug. The ideas in page 342 of the book [64] suggest the introduction of a notion of regular stability. See also [126] and [16]. Let us formulate a tentative version: Given a system ẋ = X(x) in ℝd , the equilibrium p∗ is r-stable if there exists a sequence {Mn } of invariant compact manifolds of dimension d satisfying Mn ⊂ ℝd , p∗ ∈ int(Mn ), diam(Mn ) → 0 and 𝜕Mn is a compact and connected manifold of dimension d − 1. It would be interesting to decide if the example described in Section 4.3 is not r-stable.

4.11 Bibliographical remarks | 129

As regards Section 4.5, the connection between Browder’s principle and the asymptotic stability of fixed points is discussed in [63] and [64]. Lemma 14 is a special case of a result due to Capietto, Mawhin and Zanolin; see [30]. Theorem 12 has its origin in a comment in the page 192 of [63]. In the context of periodic equations in the plane, Krasnosel’skiĭ says that a stable and isolated periodic solution has index one. By definition, this index is the fixed point index of the associated Poincaré map. This remark motivated Dancer and me to prove Theorem 12 in [38]. Later, Ruiz del Portal presented in [112] an analogous result for orientation-reversing embeddings. A challenging question is the connection between stability and index for invariant sets different from fixed or periodic points. The paper by Bell and Meyer [12] includes interesting discussions on invariant Cantor sets in the plane. In particular they proved that a stable Cantor set is never an isolated invariant set. More precisely, there exists a sequence of periodic points (with increasing periods) accumulating on the Cantor set. Ruiz-Herrera and I obtained a refinement of Bell–Meyer’s result in [101]. We proved that there exists a sequence of regions Rn accumulating on the Cantor set with corresponding iterates hσ(n) of the map h such that the fixed point index of hσ(n) in Rn is one. Example 10 appeared first in [94]. The idea of applying Yorke’s result (see [135]) was suggested to me by Dancer. Other examples have been constructed by Ruiz del Portal and Salazar in [113]. These examples are deeper because they describe all the possible configurations for the index of the iterates of the map having a stable fixed point in dimension d ≥ 3. As regards Section 4.6, sometimes the boundary of an invariant and simply connected domain is called an “invariant curve”. Then Proposition 9 implies that every perpetually stable fixed point is surrounded by “invariant curves”. This terminology is suggestive but can be misleading. Handel constructed in [55] a perpetually stable fixed point whose “invariant curves” were pseudo-circles (an intricate class of non-locally connected continua). Proposition 9 was proved in Section 42 of [14]. There Birkhoff employed the technique of filling in the holes. Given a domain Ω, he defined Ω̂ as the set of points occluded by Ω. A point P is occluded by Ω if it is possible to draw a Jordan curve lying in Ω and enclosing P. This process is also discussed in Section 25 of the book [121]. Lemma 16 is taken from [96]. The notation Γ1 ∨ Γ2 is inspired by [70]. In that paper Le Calvez finds deep connections between the index and the dynamics around a fixed point. Theorem 12 was proved in [38]. As regards Section 4.7, the first criteria for instability using nonlinear terms were obtained by Levi-Civita in his beautiful paper [75]. He analyzed the resonance at roots of the unity and also the case of a linear maps with nilpotent blocks. Proofs in [75] are based on a direct analysis of the dynamics around the fixed point. Different proofs, based on Lyapunov functions and Cetaev Theorem can be found in [121] and in [27]. The connection between degree and resonance was presented in [38]. See also [98].

130 | 4 Index of stable fixed points and periodic solutions As regards Section 4.8, global asymptotic stability is of great interest for applications and there is a very large number of papers on this topic. Sometimes the terminology is changed. For example, in the book by Pliss [106] a convergent system is a system of periodic equations with a g. a. s. periodic solution. The monograph by LaSalle [68] posed the problem of finding criteria for the existence of g. a. s fixed points based on the linearization method. This is the origin of the so-called Markus–Yamabe conjecture. See [33] for some recent results in this line. The paper by Alarcón, Guíñez and Gutiérrez [1] was also motivated by this conjecture. In that paper they introduced the condition of the invariant ray and proved the first version of Theorem 13. The condition (ii) of no escape to infinity was replaced by a stronger assumption, the embedding was dissipative. In essence this means that infinity is a repeller. The proof in [1] was based on the theory of free embeddings, a different proof based on the theory of prime ends was given in [100]. Example 9 was suggested to me by Alfonso Ruiz Herrera. Initially I constructed a more involved example. Exercise 57 is motivated by the results in [10]. The condition of the invariant ray was generalized by Ruiz-Herrera in [114], in this extension the homeomorphism can have several fixed points and there is an invariant graph connecting them, the conclusion of global attraction is replaced by trivial dynamics. Theorem 14 is a variant of a result on extinction obtained in [100]. Other applications of the theory of free embeddings to population dynamics can be found in [102, 29, 28, 114, 73]. As regards Section 4.9, it seems that, in some classical works on area-preserving maps, the equivalence between invariance and positive invariance was given for granted. See page 150 in [107] and page 80 in [14]. Lemma 17 shows that this is the case under a mild topological assumption. This lemma is essentially taken from [96]. Proposition 11 is taken from [98], as is the proof of Theorem 15. The original proof of Theorem 15 in [96] was based on Theorem 8. The local structure of analytic sets in the plane plays an important role in both proofs. We refer to [62] for more details on real analytic sets. As regards Section 4.10.1, Theorem 16 can be found in [96]. The analytic character of the Poincaré map when the vector field is not smooth in time is discussed in the book by Lefschetz [72]. Other applications of Theorem 16 can be found in [96]. They are concerned with the forced versions of the pendulum equation and Duffing’s equation. See also the paper [80] by Marò, dealing with a relativistic model. The nice trick to find a priori bounds in Ward’s paper [131] is employed in Example 10. As regards Section 4.10.2, more information on symplectic maps and Poincaré sections can be found in [121].

5 Proof of the arc translation lemma M. Brown published a short and elegant proof in 1984. We will present a more detailed version with some small differences. In fact Brown worked with homeomorphisms and we will work with embeddings. After the proof, some additional information on the location of the fixed point will be obtained.

5.1 Equivalence of embeddings Given two maps h and g in ℰ (ℝ2 ), we say that they are strongly equivalent if there is a disk D with h(D) ∩ D = 0

and h = g on ℝ2 \ D.

Sometimes D will be called a disk of modification. First of all we observe that the roles of h and g can be interchanged. Actually, h and g must coincide on 𝜕D and so h(𝜕D) = g(𝜕D). If we apply Lemma 6 in Chapter 3 with Γ = 𝜕D, we are led to h(D) = g(D) and so the conditions h(D) ∩ D = 0 and g(D) ∩ D = 0 are equivalent. Another consequence of the definition of strong equivalence is the coincidence between the sets of fixed points. The common set of fixed points will be denoted by F := Fix(h) = Fix(g). The sets F and D are closed and disjoint and so it is possible to a find a neighborhood 𝒰 of F which is also disjoint with D. Once we know that h and g coincide in a neighborhood of F we can apply the excision property and conclude that deg(id − h, Ω) = deg(id − g, Ω) for any Ω bounded and open subset of ℝ2 with 𝜕Ω ∩ F = 0. The maps h, g ∈ ℰ (ℝ2 ) are freely equivalent if there exists a chain h = h1 , h2 , . . . , hk = g in ℰ (ℝ2 ) such that hi and hi+1 are strongly equivalent for each i = 1, . . . , k − 1. The set of fixed points and the degree are also preserved under free equivalence. To get some familiarity with these notions we discuss an example. Consider a disk D = [a, b] × [0, 1] and let ϕ be a homeomorphism of D satisfying ϕ(p) = p if p ∈ 𝜕D,

ϕ(p) ≠ p if p ∈ int(D).

For instance ϕ could be the time map of a flow on D whose equilibria are precisely the points in 𝜕D. See Figure 5.1. https://doi.org/10.1515/9783110551167-005

132 | 5 Proof of the arc translation lemma

Figure 5.1: A flow inducing ϕ.

The translation T(x, y) = (x + 1, y) will be modified to the map TD in ℰ (ℝ2 ) defined by TD = T ∘ ϕ

on D,

TD = T

outside D.

If b − a < 1 the maps T and TD are strongly equivalent with D as a disk of modification. It is interesting to observe that T and TD are not strongly equivalent if b − a ≥ 1. This is a consequence of the following. Exercise 65. Given h, g ∈ ℰ (ℝ2 ), the set 𝒩 is the closure of {p ∈ ℝ2 : h(p) ≠ g(p)}. Prove that if h(𝒩 ) ∩ 𝒩 ≠ 0 then h and g are not strongly equivalent. Next we consider the disks 1 D1 = [0, ] × [0, 1], 2

1 D2 = [ , 1] × [0, 1] 2

and corresponding homeomorphisms ϕ1 and ϕ2 as before. A new map T̂ is defined by T̂ = T ∘ ϕ1

on D1 ,

T̂ = T ∘ ϕ2

on D2 ,

T̂ = T

outside D1 ∪ D2 .

The chain h1 = T, h2 = TD1 , h3 = T̂ shows that T and T̂ are freely equivalent. However, in view of Exercise 65, they are not strongly equivalent.

5.2 Compression of translation arcs In this section we show how to contract a translation arc via free modifications. ̂ Throughout the section α = p q is an arc and p⋆ is a given point in α̇ = α \ {p, q}. This ⋆ q. ̂⋆ and α󸀠󸀠 = p̂ point splits the arc α in two sub-arcs α󸀠 = pp Proposition 12. Assume that α is a translation arc for some h ∈ ℰ (ℝ2 ) and 𝒰 is a neighborhood of α󸀠 . Then there exists g ∈ ℰ (ℝ2 ) which is strongly equivalent to h with disk of

5.2 Compression of translation arcs | 133

modification contained in 𝒰 and such that g(α󸀠󸀠 ) = h(α),

g(p⋆ ) = q,

g(q) = h(q).

See Figure 5.2.

Figure 5.2: From α to α 󸀠󸀠 .

Notice that α󸀠󸀠 is a translation arc for g. Before proving this proposition we state a simpler result which seems obvious. Lemma 19. Let D be a disk with α󸀠 ⊂ int(D). Then there exists ϕ ∈ ℋ(ℝ2 ) with ϕ(p) = p⋆ ,

ϕ(q) = q,

ϕ(α) = α󸀠󸀠 ,

ϕ = id outside D.

Proof. We use the fact that arcs are tamely embedded in the plane. This means that we can find ψ ∈ ℋ(ℝ2 ) with ψ(α) = [0, 1] × {0},

ψ(p) = (0, 0),

1 ψ(p⋆ ) = ( , 0), 2

ψ(q) = (1, 0).

For the special arc [0, 1] × {0} it is easy to find an explicit homeomorphism ϕ0 under the conditions required by the Lemma and such that ϕ0 = id outside the disk ψ(D). Now we can go back to α with ϕ = ψ−1 ∘ ϕ0 ∘ ψ. See Figure 5.3.

Figure 5.3: The arc can be rectified.

134 | 5 Proof of the arc translation lemma Exercise 66. Construct ϕ0 . Proof of Proposition 12. Since α is a translation arc for h, we know that h(α󸀠 ) ∩ α󸀠 = 0. This allows us to find a disk D with α󸀠 ⊂ int(D), D ⊂ 𝒰 and h(D) ∩ D = 0. This disk can be obtained by inflating α󸀠 . The previous lemma produces a homeomorphism ϕ contracting α to α󸀠󸀠 . The embedding g = h ∘ ϕ−1 is strongly equivalent to h with D as a disk of modification. In the next result we show how to compress the image of a translation arc. At first sight it seems that such a result could be obtained by applying Proposition 12 to h−1 . We must take some care since we are dealing with embeddings and not with homeomorphisms. Proposition 13. Assume that α is a translation arc for some h ∈ ℰ (ℝ2 ) and h(q) ∈ ̸ α. Given 𝒱 neighborhood of α󸀠󸀠 there exists g ∈ ℰ (ℝ2 ) strongly equivalent to h, with a disk of modification contained in 𝒱 and such that g(α) = h(α󸀠 ),

g(p) = q,

g(q) = h(p⋆ ).

See Figure 5.4.

Figure 5.4: From h(α) to h(α 󸀠 ).

Proof. The basic properties of embeddings which were discussed at the beginning of Chapter 3 imply that, if Δ is a disk with α󸀠󸀠 ⊂ int(Δ), Δ ⊂ 𝒱 , then h(Δ) = D is also a disk with h(α󸀠󸀠 ) ⊂ int(D). Since h(q) ∈ ̸ α, the arcs α󸀠󸀠 and h(α󸀠󸀠 ) are disjoint. We can select D small enough so that D ∩ Δ = 0. We apply Lemma 19 and find ϕ compressing h(α) onto h(α󸀠 ) and such that ϕ = id outside D. The searched map is g = ϕ ∘ h and Δ is the disk of modification.

5.3 Reduction to periodic orbits ̂ Assume that α = p q is a translation arc for h ∈ ℰ (ℝ2 ). We are going to associate an index ν = ν(α) to this arc. It takes the values ν = 2, 3, . . . , ∞ according to

5.3 Reduction to periodic orbits | 135

Figure 5.5: Two cases of index ν = 5.

– –

ν = 2 whenever h(q) ∈ α; 3 ≤ ν < ∞ whenever h(q) ∈ ̸ α and hk (α) ∩ α = 0 if 2 ≤ k < ν − 1,

–

hν−1 (α) ∩ α ≠ 0;

ν = ∞ whenever hk (α) ∩ α = 0, k ≥ 2.

Notice that the condition h(q) ∈ α implies that h2 (α) ∩ α ≠ 0. Exercise 67. Compute ν in the following cases: h(x, y) = (x + 1, y),

h(z) = eiΘ z,

0 < Θ ≤ π,

α = [0, 1] × {0},

α = {eiθ : 0 ≤ θ ≤ Θ}.

̂ Exercise 68. Assume that α = p q is a translation arc with ∞ > ν ≥ 3. Then q ∈ ̸ hν−1 (α). In Figure 5.5 we illustrate two possible situations having the same index. We will show that, after a free modification, we have reduction to the periodic case. Proposition 14. Assume that α is a translation arc for h ∈ ℰ (ℝ2 ) and ν < ∞. Then there exist g ∈ ℰ (ℝ2 ), which is freely equivalent to h, a periodic orbit with minimum period ν, P0 ,

P1 = g(P0 ),

...,

Pν = g ν (P0 ) = P0 ,

and a translation arc for g denoted by β = P̂ 0 P1 such that Γ = β ∪ g(β) ∪ ⋅ ⋅ ⋅ ∪ g ν−1 (β) is a Jordan curve contained in α ∪ h(α) ∪ ⋅ ⋅ ⋅ ∪ hν−1 (α). We first state a preliminary result whose proof is immediate. Lemma 20. Assume that a0 , a1 , . . . , ar−1 are r ≥ 3 different points which are connected by arcs γ0 = â 0 a1 ,

γ1 = â 1 a2 ,

...,

γr−1 = a? r−1 a0

136 | 5 Proof of the arc translation lemma satisfying γ0 ∩ γ1 = {a1 },

...,

γr−2 ∩ γr−1 = {ar−1 },

γr−1 ∩ γ0 = {a0 }

and γj ∩ γk = 0

if 2 ≤ |j − k| < r − 1.

Then Γ = γ0 ∪ γ1 ∪ ⋅ ⋅ ⋅ ∪ γr−1 is a Jordan curve. Proof of Proposition 14. Assume first that ν = 2 so that the point p⋆ = h(q) lies in α\{q}. If p⋆ = p there is no need to modify h, since {p, q} is a periodic orbit and α ∪ h(α) is a Jordan curve. Assume now that p⋆ ∈ α,̇ then Γ = α󸀠󸀠 ∪ h(α) is a Jordan curve with ⋆ q. We apply Proposition 12 and compress α to α󸀠󸀠 . In this way we obtain g, α󸀠󸀠 = p̂ strongly equivalent to h and having the periodic orbit {p⋆ , q} and the translation arc β = α󸀠󸀠 . Assume from now on that 3 ≤ ν < ∞. We define the arcs α0 = α,

α1 = h(α),

αν−1 = hν−1 (α)

...,

and observe that they satisfy αj ∩ αk = 0 if 2 ≤ |j − k| < ν − 1 and α0 ∩ α1 = {h(p)},

α1 ∩ α2 = {h2 (p)},

...,

αν−2 ∩ αν−1 = {hν−1 (p)}.

This is a consequence of the definition of ν if we use the fact that h is one-to-one and that α is a translation arc. The arcs α0 , α1 , . . . , αν−1 are almost under the conditions of Lemma 20, excepting that α0 and αν−1 can intersect many times. In view of Exercise 68 the point q is excluded from these intersections. We select p⋆ as the first point in the arc αν−1 which encounters α0 . See Figure 5.6. In principle, the point p⋆ can be anywhere on α\{q}. In particular it could coincide with p. From the definition of ν we know that hν−1 (p) = hν−2 (q) cannot belong to α and therefore p∗ must lie in αν−1 \ {hν−1 (p)}. From now on we assume that p⋆ ≠ p, hν (p). This assumption contains the most difficult cases. Consider the sub-arc of α from p⋆ to q, ⋆q α0󸀠󸀠 = p̂

5.3 Reduction to periodic orbits | 137

Figure 5.6: p⋆ is the first point in αν−1 lying also in α.

and the sub-arc of αν−1 from hν−1 (p) to p⋆ , 󸀠 ? = hν−1 (p)p⋆ . αν−1 󸀠 By construction, the arcs α0󸀠󸀠 , α1 , . . . , αν−2 , αν−1 are under the conditions of Lemma 20 and 󸀠 Γ = α0󸀠󸀠 ∪ α1 ∪ ⋅ ⋅ ⋅ ∪ αν−2 ∪ αν−1

is a Jordan curve. The successive images of α are also translation arcs for h. In particular, αν−2 is a translation arc with end points hν−2 (p) and hν−2 (q). Since ν > 2 we know that h(q) ∈ ̸ α and this implies that h(hν−2 (q)) ∈ ̸ αν−2 . We are now in the position to apply ? ν−2 (p)r, α󸀠󸀠 = rh ν−1 (p), Proposition 13 with α = α . We employ the notations α󸀠 = h? ν−2

ν−2

ν−2

where r is such that h(r) = p⋆ . Note that r ∈ α̇ ν−2 . In this way we obtain g1 ∈ ℰ (ℝ2 ) strongly equivalent to h and such that 󸀠 . g1 (αν−2 ) = αν−1

󸀠󸀠 The disk of modification has to contain αν−2 in its interior but we can select it small enough so that it does not intersect α0 ∪ α1 ∪ ⋅ ⋅ ⋅ ∪ αν−3 . The maps h and g1 coincide in these arcs and so

g1 (α0 ) = α1 ,

...,

g1 (αν−3 ) = αν−2 .

The arc α is also a translation arc for g1 and we can apply Proposition 12 and obtain g ∈ ℰ (ℝ2 ) strongly equivalent to g1 and such that g(α0󸀠󸀠 ) = α1 . The disk of modification is chosen in a neighborhood of α0󸀠 which does not intersect α1 ∪ ⋅ ⋅ ⋅ ∪ αν−2 . This leads to g(α1 ) = α2 ,

...,

󸀠 g(αν−2 ) = αν−1 .

This completes the proof of the proposition under the assumption p⋆ ≠ p, hν (p).

138 | 5 Proof of the arc translation lemma Exercise 69. Adapt the previous proof to the cases p⋆ = p ≠ hν (p) and p⋆ = hν (p) ≠ p. What can be said about the case p⋆ = p = hν (p)? A consequence of Proposition 14 is that the set of fixed points F = Fix(g) = Fix(h) n is disjoint with Γ or even with ⋃∞ n=0 h (α). This follows easily from Exercise 70. Assume that X is a set and f : X → X is a one-to-one mapping. If A is a subset of X without fixed points then f n (A) ∩ Fix(f ) = 0 for each n = 0, 1, 2, . . . The maps h and g are freely equivalent and this implies that deg(id − h, Ri (Γ)) = deg(id − g, Ri (Γ)). The curve Γ is not always invariant under g. In fact g(Γ) = g(β) ∪ ⋅ ⋅ ⋅ ∪ g ν−1 (β) ∪ g ν (β) and we cannot guarantee that g ν (β) lies in Γ. At this moment it is convenient to present a short summary of the next steps towards the proof of the arc translation lemma. The idea will be to construct a continuous deformation of g, {gt }t∈[0,1] , in such a way that g0 = g and Γ is invariant under g1 . See Figure 5.7. This will be done in such a way that id − g and id − g1 are homotopic on Ri (Γ) and so deg(id − g, Ri (Γ)) = deg(id − g1 , Ri (Γ)). Then we will apply the result on the degree inside an invariant curve which is stated and proved in the appendix. In the next section we develop the tools to construct the deformation {gt }.

Figure 5.7: A deformation towards the invariant curve.

5.4 Lemmas on isotopies | 139

5.4 Lemmas on isotopies The exterior of the unit disk is denoted by 𝔼 = {p ∈ ℝ2 : ‖p‖ ≥ 1}. The boundary of 𝔼 is the unit circle 𝜕𝔼 = 𝕊1 . The class of mappings h : 𝔼 → 𝔼 which are continuous and one-to-one will be denoted by ℰ (𝔼). Two maps h0 , h1 ∈ ℰ (𝔼) are isotopic relative to 𝜕𝔼 if h0 = h1 on 𝜕𝔼 and there exists a continuous map H : [0, 1] × 𝔼 → 𝔼,

(t, p) 󳨃→ Ht (p)

such that – H0 = h0 , H1 = h1 ; – Ht ∈ ℰ (𝔼) for each t ∈ [0, 1]; – Ht (p) = h0 (p) = h1 (p) if p ∈ 𝜕𝔼 and t ∈ [0, 1]. Lemma 21 (Alexander’s isotopy). Assume that h ∈ ℰ (𝔼) and h = id on 𝜕𝔼. Then h is isotopic to id relative to 𝜕𝔼. Proof. Define 1

Ht (p) = { t p

h(tp) if t‖p‖ ≥ 1, if t‖p‖ ≤ 1.

To check that the map H satisfies the required properties is more or less automatic. It is perhaps more interesting to understand this isotopy geometrically. For each t ∈ ]0, 1] we distinguish the sets 1 At = {p ∈ 𝔼 : 1 ≤ ‖p‖ ≤ }, t

1 Bt = {p ∈ 𝔼 : ‖p‖ ≥ } t

and observe that Ht = id on At and Ht = D−1 t ∘ h ∘ Dt on Bt . Here Dt is the linear contraction Dt (p) = tp. As the time t grows from 0 to 1 the annulus At shrinks until collapsing on 𝜕𝔼. See Figure 5.8.

Figure 5.8: The isotopy H.

140 | 5 Proof of the arc translation lemma Exercise 71. Let S(x, y) = (x, −y) be the orthogonal symmetry with respect to the x axis. Assume that h ∈ ℰ (𝔼) and h = S on 𝜕𝔼. Then h is isotopic to S relative to 𝜕𝔼. We go back to the whole plane ℝ2 and employ the notion of isotopy in ℰ (ℝ2 ) relative to an arc α. The reader can state the precise definition. Lemma 22. Let α be an arc in ℝ2 and h ∈ ℰ⋆ (ℝ2 ) satisfies h = id on α. Then h is isotopic to id relative to α. It is important to observe that we are assuming that h is orientation-preserving. Without this assumption the lemma is false, as shown by the example h(x, y) = (x, −y),

α = [0, 1] × {0}.

We justify this assertion by a contradiction argument. If Ht were an isotopy from h to id relative to α, then Ht (0, 0) = (0, 0) for each t and the homotopy property of the degree would imply that deg(h, 𝒰 ) = deg(id, 𝒰 ) on any open and bounded neighborhood of the origin. This is absurd since deg(h, 𝒰 ) = −1. Lemma 22 is, according to Brown, a “folk theorem”. Brown sketched the proof as follows: “cut open the arc α to form a disk and apply the Alexander isotopy to the exterior of the disk”. Next we provide the details of this procedure. Proof of Lemma 22. First of all we take ϕ ∈ ℋ(ℝ2 ) mapping α onto the segment β = [0, 1] × {0}. It is enough to prove the result for g = ϕ ∘ h ∘ ϕ−1 and the arc β. Notice that the invariance of the degree under changes of variable implies that g ∈ ℰ∗ (ℝ). We cut the segment to form a disk. This means that we are going to distinguish the two sides of β in ℝ2 \ β. To this end we consider the topological space ̇ X = (ℝ2 \ β)̇ ∪ {(r, +) : r ∈ β}̇ ∪ {(r, −) : r ∈ β}. The points p⋆ = (0, 0), q⋆ = (1, 0) are the end points of β. The definition of the topology in X is natural if it must produce the continuity of the projection π : X → ℝ2 and to make X homeomorphic to 𝔼. See Figure 5.9. We just notice that a sequence {pn } in ℝ2 \ β converges to (r, +) in X if and only if pn → r and yn > 0 for n large enough, where pn = (xn , yn ).

Figure 5.9: An open cut of the arc.

5.4 Lemmas on isotopies | 141

Exercise 72. Define a distance on X inducing this convergence. Given r ∈ β,̇ we consider the disk D = {p ∈ ℝ2 : ‖p − r‖ ≤ ϵ}. We can choose ϵ small enough so that g(D) is contained in the vertical strip β × ℝ. This is just a consequence of the continuity of g at r. Define H + = {(x, y) ∈ ℝ2 : y > 0},

H − = {(x, y) ∈ ℝ2 : y < 0}

D+ = D ∩ H + ,

D− = D ∩ H − .

We claim that one of the possibilities below holds: (i)

g(D+ ) ⊂ H + ,

(ii) g(D+ ) ⊂ H − ,

g(D− ) ⊂ H − ,

g(D− ) ⊂ H + .

See Figure 5.10.

Figure 5.10: The orientation-reversing case.

To prove this claim we first notice that g(D+ ) and g(D− ) cannot intersect the axis y = 0. This is a consequence of g(D) ⊂ β × ℝ and g = id on β × {0}. The sets g(D+ ) and g(D− ) are connected and contained in H + ∪ H − . Each of them must be either in H + or in H − . It remains to prove that both of them cannot lie in the same half-plane. To this end we proceed by contradiction. If g(D+ ) and g(D− ) were in the same half-plane, then r should belong to the boundary of the disk g(D). But g is an open map and so r ∈ int(D) should imply r = g(r) ∈ g(int(D)) = int(g(D)). Then r would be simultaneously on the boundary and in the interior of the disk g(D) and this is the searched contradiction. In principle, the alternative (i)–(ii) could change with a different choice of the disk D. We show that this is not the case. Given two disks D1 and D2 , centered at p and such that g(D1 ) ∪ g(D2 ) ⊂ β × ℝ, we know that D+1 ∩ D+2 ≠ 0. In consequence, g(D+1 ) ∩ g(D+2 ) ≠ 0 and the same alternative, (i) or (ii), will hold for the two disks. Another useful remark is that the validity (i) or (ii) is common to all the points in β.̇ To prove this assertion we consider the sets A = {r ∈ β̇ : (i) holds},

B = {r ∈ β̇ : (ii) holds}.

142 | 5 Proof of the arc translation lemma Both sets are open in β̇ and they satisfy A ∪ B = β,̇ A ∩ B = 0. Since β̇ is connected, one of them is empty. After this discussion of the alternative (i)–(ii), it is possible to define a continuous and one-to-one map ĝ : X → X ̂ ±) = (r, ±) in the case (i), satisfying π ∘ ĝ = g ∘ π. Namely, g = ĝ on ℝ2 \ β̇ and g(r, ̂ ±) = (r, ∓) in case (ii). g(r, In the space X we consider the arcs C± = {(r, ±) : r ∈ β}̇ ∪ {p⋆ , q⋆ }. The Jordan curve Γ = C+ ∪ C− is invariant under g.̂ In case (i) all the points in Γ are fixed. Since the pairs (𝔼, 𝕊1 ) and (X, Γ) are homeomorphic, we can apply Lemma 21 and obtain an isotopy Ĥ t : X → X between id and g.̂ This isotopy is relative to Γ and so it induces an isotopy Ht in ℝ2 between id and g. Indeed, Ht = Ĥ t

on ℝ2 \ β,̇

Ht = id

on β.

This argument completes the proof in case (i). It remains to prove that the case (ii) cannot hold and it is here where we shall use the fact that h and g = ϕ ∘ h ∘ ϕ−1 are orientation-preserving. We first notice that the symmetry S(x, y) = (x, −y) induces a homeomorphism of X which will be denoted by S.̂ It satisfies S ∘ π = π ∘ S.̂ If g is in case (ii) then Ŝ ∘ ĝ = id on Γ. We apply again Lemma 21, but this time to Ŝ ∘ g,̂ and find an isotopy ℋ̂ t : X → X between id and Ŝ ∘ g,̂ relative to Γ. It induces an isotopy ℋt : ℝ2 → ℝ2 between id and S ∘ g which is relative to β. Given any open and bounded neighborhood 𝒰 of the origin we observe that deg(S ∘ ℋt , 𝒰 ) is independent of t. This is so because (S ∘ ℋt )−1 (0) = 0 and one can apply the invariance by homotopies. In particular we observe that S ∘ ℋ1 = g, S ∘ ℋ0 = S and so deg(g, 𝒰 ) = deg(S, 𝒰 ) = −1. This is excluded since g belongs to ℰ⋆ (ℝ2 ).

5.5 Computation of the degree on certain Jordan domains The goal of this section is to prove the following result. Proposition 15. Assume that h ∈ ℰ⋆ (ℝ2 ) and γ is an arc in ℝ2 such that Fix(h) ∩ γ = 0 and Γ = γ ∪ h(γ) is a Jordan curve. Then Fix(h) ∩ Γ = 0 and deg(id − h, Ri (Γ)) = 1.

5.5 Computation of the degree on certain Jordan domains |

143

Up to now arcs have been denoted by α and now we have changed to γ. The intention is to indicate that γ is not necessarily a translation arc. For instance, if h is a rotation of 90 degrees, then any arc γ = {eiθ : θ ∈ [0, Θ]} with 2π > Θ ≥ 3π is in the 2 conditions of the proposition but it is not a translation arc. Given h ∈ ℰ (ℝ2 ), we say that Γ ⊂ ℝ2 is an invariant curve if it is a Jordan curve satisfying h(Γ) = Γ. We say that the invariant curve has no fixed points if Fix(h) ∩ Γ = 0. Before proving Proposition 15 we need to present some properties of invariant curves. Lemma 23. Assume that h ∈ ℰ (ℝ2 ) and Γ is an invariant curve without fixed points. Then deg(id − h, Ri (Γ)) = 1. This result is a consequence of Proposition 16 in the appendix. In the previous lemma we did not impose the preservation of orientation for h, however, we will prove that this is a consequence of the assumptions. Lemma 24. Assume that h ∈ ℰ (ℝ2 ) has an invariant curve without fixed points. Then h ∈ ℰ⋆ (ℝ2 ). To prepare for the proof of this lemma we recall some facts about homeomorphisms of the circle. Given k ∈ ℋ(𝕊1 ) there exists a lift k̃ : ℝ → ℝ such that k(eiθ ) = eik(θ) ̃

for each θ ∈ ℝ.

The lift is a homeomorphism of the real line and it satisfies ̃ + 2π) = k(θ) ̃ ± 2π, k(θ where the + sign appears for increasing k̃ and the − sign for decreasing k.̃ If we assume that k has no fixed points then ̃ − θ − 2kπ ≠ 0 k(θ) for each θ ∈ ℝ and k ∈ ℤ. This is impossible if k̃ is decreasing and so we conclude that k̃ is increasing if Fix(k) = 0. Proof of Lemma 24. Assume first that Γ = 𝕊1 . The restriction of h to 𝕊1 produces a map k ∈ ℋ(𝕊1 ). By assumption, it has no fixed points and so h(eiθ ) = eik(θ) ̃

with k̃ : ℝ → ℝ an increasing homeomorphism. Consider the map in the plane ̃ ̂ iθ ) = reik(θ) h(re .

144 | 5 Proof of the arc translation lemma It belongs to ℋ(ℝ2 ) and satisfies h = ĥ on 𝕊1 . In particular, deg(h, int(𝔻)) = deg(h,̂ int(𝔻)), where 𝔻 is the unit disk. We construct an isotopy between ĥ and id which is easily expressed in polar coordinates, Ht :

{

̃ θ1 = tθ + (1 − t)k(θ), r1 = r.

The map Ht is a homeomorphism because k̃ is increasing. We also observe that H0 = h,̂ H1 = id and Ht−1 (0) = 0 for each t. The homotopy property implies that deg(h,̂ int(𝔻)) = deg(id, int(𝔻)) = 1. Finally, we consider the homotopy H(p, λ) = h(p) − λh(0),

λ ∈ [0, 1].

If p ∈ 𝕊1 = 𝜕𝔻, ‖H(p, λ)‖ ≥ ‖h(p)‖−λ‖h(0)‖ ≥ 1−‖h(0)‖ > 0. Then deg(h−h(0), int(𝔻)) = deg(h, int(𝔻)) = 1. This completes the proof for Γ = 𝕊1 . Assume now that Γ is an arbitrary Jordan curve. Then we can find ϕ ∈ ℋ(ℝ2 ) with ϕ(Γ) = 𝕊1 . The unit circle becomes an invariant curve without fixed points for the map ϕ ∘ h ∘ ϕ−1 . The previous discussions imply that ϕ ∘ h ∘ ϕ−1 is in ℰ⋆ (ℝ2 ) and this property is transferred to h by the invariance of the degree under changes of variable. Proof of Proposition 15. This result would be a consequence of Lemma 23 if Γ were invariant. In general Γ will not be invariant since h(γ) can be mapped outside Γ. We construct a homeomorphism of Γ, say k : Γ → Γ, such that k = h on γ. To do this we have some freedom. The arc σ = Γ \ γ̇ is mapped homeomorphically onto ̇ The end points must be transformed appropriately so that the resulting k σ1 = Γ \ h(γ). is continuous. The map k is a homeomorphism of Γ and now we are going to prove that it has no fixed points. We apply Exercise 70 with A = γ and X = Γ. Then k has no fixed point on γ ∪ h(γ) = Γ. Next we apply the Jordan–Schönflies theorem to extend k to a homeomorphism ϕ ∈ ℋ(ℝ2 ). The curve Γ is invariant under ϕ and fixed point free. We can apply Lemma 24 to deduce that ϕ, ϕ−1 ∈ ℰ⋆ (ℝ2 ). From Exercise 26 we deduce that ϕ−1 ∘ h ∈ ℰ⋆ (ℝ2 ) and it is at this point that we have used the assumption h ∈ ℰ⋆ (ℝ2 ). The map ϕ−1 ∘h coincides with the identity on γ and we can apply Lemma 22 with α = γ and h = ϕ−1 ∘ h. There exists an isotopy {Gt } with G0 = ϕ−1 ∘ h, G1 = id, Gt = id on γ. The family of maps Ht = ϕ ∘ Gt defines an isotopy between h and ϕ. Since Ht = ϕ = h on γ,

5.6 Brouwer’s lemma and some consequences |

145

we know that Ht has no fixed points in γ. Once again we apply Exercise 70 to deduce that Ht has no fixed points on γ ∪ Ht (γ) = γ ∪ ϕ(γ) = γ ∪ k(γ) = Γ. In particular, deg(id − h, Ri (Γ)) = deg(id − ϕ, Ri (Γ)). The conclusion follows after applying Lemma 23 to ϕ.

5.6 Brouwer’s lemma and some consequences Let us prove Brouwer’s lemma as stated in Chapter 3. We are given h ∈ ℰ⋆ (ℝ2 ) and a translation arc α with hn (α) ∩ α ≠ 0 for some n ≥ 2. This means that the index of this arc is finite, say ν ≥ 2. We apply Proposition 14 and find g ∈ ℰ (ℝ2 ), freely equivalent to h, with a periodic orbit P0 , P1 , . . . , Pν−1 and a translation arc β = P̂ 0 P1 such that Γ = β ∪ g(β) ∪ ⋅ ⋅ ⋅ ∪ g ν−1 (β) is a Jordan curve contained in α ∪ h(α) ∪ ⋅ ⋅ ⋅ ∪ hν−1 (α). We notice that the equivalence of h and g implies that they coincide outside a compact set and so also g belongs to ℰ⋆ (ℝ2 ). We are using Exercise 24 of Chapter 3. Next we shall apply Proposition 15 to the map g and the arc γ = β ∪ g(β) ∪ ⋅ ⋅ ⋅ ∪ g ν−2 (β). We observe that Fix(g) ∩ γ = 0 because γ is a translation arc. Also, γ ∪ g(γ) = Γ is a Jordan curve and so we conclude that deg(id − g, Ri (Γ)) = 1. The properties of equivalent embeddings imply that deg(id − g, Ri (Γ)) = deg(id − h, Ri (Γ)) and this completes the proof. The previous proof provides extra information about the curve Γ. It satisfies Γ ⊂ α ∪ h(α) ∪ ⋅ ⋅ ⋅ ∪ hν−1 (α). This fact can be useful for the location of fixed points. Theorem 18. Assume that h ∈ ℰ⋆ (ℝ2 ) and α is a translation arc with finite index ν ≥ 2. Then h has a fixed point in the some bounded component of ℝ2 \ (α ∪ h(α) ∪ ⋅ ⋅ ⋅ ∪ hν−1 (α)). Proof. We know that h has a fixed point p⋆ in Ri (Γ) with Γ ⊂ α∪h(α)∪⋅ ⋅ ⋅∪hν−1 (α). Since α is a translation arc, the point p∗ cannot lie on α ∪ h(α) ∪ ⋅ ⋅ ⋅ ∪ hν−1 (α). We will prove that p⋆ belongs to some bounded component of ℝ2 \ [α ∪ h(α) ∪ ⋅ ⋅ ⋅ ∪ hν−1 (α)]. Otherwise p⋆ and ∞ should lie in the same connected component of 𝕊2 \ [α ∪ h(α) ∪ ⋅ ⋅ ⋅ ∪ hν−1 (α)], say Ω. Here we are using the Riemann sphere 𝕊2 = ℝ2 ∪ {∞}. The connected set Ω is contained in 𝕊2 \ Γ and so, by Jordan’s theorem, p⋆ should lie in Re (Γ). This is absurd since we started with a fixed point p⋆ lying in Ri (Γ).

5.7 Bibliographical remarks As regards Section 5.1, the notion of free equivalence for homeomorphisms was introduced in [23]. The terminology was probably inspired by knot theory. As regards Section 5.2, Lemma 19 corresponds to Lemma 2 in [23]. For more information on tame sets in the plane the reader is referred to the book [87].

146 | 5 Proof of the arc translation lemma As regards Section 5.3, the proof of Proposition 14 is extracted from the second case in the proof of Theorem 1 in [23]. Exercise 70 is inspired by Lemma 3 in [23]. As regards Section 5.4, a homeomorphism of ℝd is called stable if it coincides with the identity on a non-empty open set. Traditionally, Alexander’s isotopy is used to prove that stable homeomorphisms are isotopic to the identity (see Chapter 11 in [87]). Lemma 1 in [23] can be seen as a variant of this result for the plane. Now the homeomorphism only coincides with the identity on an arc but we know in advance that it is orientation-preserving. As pointed out by Brown in [23], the use of Alexander’s isotopy is a crucial step in his proof on the arc translation lemma. In previous proofs the property of orientation-preserving was used in rather intuitive arguments. Lemma 22 is just an extension of Lemma 1 in [23] to embeddings. The construction of the space X is one of the many ways to give a precise construction according to the recipe: cut open a segment. The reader who is familiar with Caratheodory’s theory of prime ends (see [109]) may consider the simply connected domain Ω = 𝕊2 \ β and the corresponding space of prime ends ℙ(Ω). Then X is homeomorphic to Ω̂ = Ω ∪ ℙ(Ω). As regards Section 5.5, Proposition 15 is a fixed point theorem inspired by the proof of the first case of Theorem 1 in [23]. More information on the lift of a homeomorphism of 𝕊1 can be found in the book [8]. The main result of this section has some formal contact with the fixed point theorem obtained by Bonino in [17]. As regards Section 5.6, Theorem 18 has been employed by Graff and Ruiz-Herrera in [49] to locate fixed points. The idea is to impose some geometric conditions on the embedding in order to find a safety region containing the translation arc and its iterates.

6 Appendix on degree theory The topological degree and the fixed point index have appeared many times in the previous pages. Now we collect all the properties that have been used. In most cases there are no proofs; the reader is referred to the section on bibliographical remarks for more information.

6.1 Definition Given an open and bounded subset Ω of ℝd , we consider continuous maps f : cl(Ω) → ℝd which do not vanish on the boundary; that is, f (x) ≠ 0

for each x ∈ 𝜕Ω.

This class of maps will be denoted by ℳ⋆ (Ω). Exercise 73. Construct two open and bounded sets Ω1 and Ω2 such that cl(Ω1 ) = cl(Ω2 ) and ℳ⋆ (Ω1 ) ≠ ℳ⋆ (Ω2 ). Given a map f ∈ ℳ⋆ (Ω), an integer number can be associated to it. This number is the degree and it will be denoted by deg(f , Ω). An important property of the degree is the invariance under continuous deformations inside the class ℳ⋆ (Ω). For instance, if d = 1 and Ω = ]a, b[, two continuous functions f , g : [a, b] → ℝ with f (a) ⋅ g(a) > 0 and f (b) ⋅ g(b) > 0 will have the same degree. Note that we can produce a continuous deformation from the graph of f to the graph of g without passing by the points (a, 0) and (b, 0). See Figure 6.1.

Figure 6.1: Continuous deformation-preserving zeros.

In precise terms this intuitive statement becomes the homotopy property: Assume that H : cl(Ω) × [0, 1] → ℝd , (x, λ) 󳨃→ H(x, λ) is continuous and such that H(x, λ) ≠ 0

for each x ∈ 𝜕Ω and λ ∈ [0, 1].

Then Hλ = H(⋅, λ) ∈ ℳ⋆ (Ω) and deg(Hλ , Ω) is independent of λ ∈ [0, 1]. https://doi.org/10.1515/9783110551167-006

148 | 6 Appendix on degree theory Sometimes it is convenient to split the domain Ω in order to compute the degree. This is possible by the additivity property: Assume that Ω, Ω1 , Ω2 ⊂ ℝd are three bounded and open sets with Ω1 ∪ Ω2 ⊂ Ω

and

Ω1 ∩ Ω2 = 0

and let f ∈ ℳ⋆ (Ω) be such that f (x) ≠ 0

for each x ∈ Ω \ (Ω1 ∪ Ω2 ).

Then for each i = 1, 2 the corresponding restrictions of f belong to ℳ⋆ (Ωi ) and deg(f , Ω) = deg(f , Ω1 ) + deg(f , Ω2 ). This property is also valid when some Ωi is empty, with the convention deg(f , 0) = 0. In particular, for the choice Ω1 = Ω2 = 0 we deduce that the degree vanishes for maps without zeros. In other words, deg(f , Ω) = 0 for each f ∈ ℳ⋆ (Ω) with f (x) ≠ 0 in Ω. Another convention of degree theory is that the identity map has degree one as soon as it has a zero. This is sometimes called the normalization property: 1 deg(id, Ω) = { 0

if 0 ∈ Ω,

if 0 ∈ ̸ cl(Ω).

These three properties characterize the degree and they can be taken as the axioms of degree theory.

6.2 Some useful properties From the previous axioms it is possible to deduce many useful consequences. Next we list some of them. The additivity property with Ω1 ≠ 0, Ω2 = 0 leads to the excision property: Assume that Ω1 ⊂ Ω and all the zeros of f ∈ ℳ⋆ (Ω) lie on Ω1 , then deg(f , Ω) = deg(f , Ω1 ). Two important consequences of the homotopy property are: – The degree only depends of the values of the map on the boundary. Given f , g ∈ ℳ⋆ (Ω) with f (x) = g(x) if x ∈ 𝜕Ω, then deg(f , Ω) = deg(g, Ω).

6.2 Some useful properties | 149

–

The degree is stable under small perturbations. Given f ∈ ℳ⋆ (Ω), there exists ϵ > 0 such that if g ∈ ℳ⋆ (Ω) and ‖f (x) − g(x)‖ < ϵ

for each x ∈ 𝜕Ω,

then deg(f , Ω) = deg(g, Ω). Exercise 74. Deduce these properties from the axioms. Many applications of the degree are related to solving equations. This is explained by the following property: – Maps with non-vanishing degree have essential zeros. Indeed, if f ∈ ℳ⋆ (Ω) and deg(f , Ω) ≠ 0, then there exists ϵ > 0 such that the equation f (x) = y,

x ∈ Ω,

has at least one solution if y ∈ ℝd with ‖y‖ < ϵ. To prove this claim we define g(x) = f (x) − y and observe that deg(g, Ω) = deg(f , Ω) ≠ 0 if ϵ is small enough. Then g must have a zero on Ω by the additivity property. The product theorem of Leray is concerned with the degree of the composition of two maps. We refer to [110] for the precise statement but we will list four consequences of this theorem that will be useful for us. – The degree of one-to-one maps. Assume that f ∈ ℳ⋆ (Ω) is one-to-one, then ±1 if 0 ∈ f (Ω), deg(f , Ω) = { 0 if 0 ∈ ̸ f (cl(Ω)). –

The degree of the second iterate (f 2 = f ∘ f ). Assume that f : ℝd → ℝd is continuous and one-to-one and Ω is connected and such that 0 ∈ f (Ω) ∩ f 2 (Ω).

–

Then deg(f 2 , Ω) = 1. The degree of a composition. Assume that Ω is connected, f : cl(Ω) → ℝd is continuous and one-to-one and g : f (cl(Ω)) → ℝd is continuous. Let us pick up a point x0 ∈ Ω and define y0 = f (x0 ), z0 = g(y0 ). Assume that g(f (x)) = z0 ,

x ∈ 𝜕Ω,

150 | 6 Appendix on degree theory has no solution. Then deg(g ∘ f − z0 , Ω) = deg(g − z0 , f (Ω)) ⋅ deg(f − y0 , Ω).

–

It must be noticed that f (Ω) is open and bounded and 𝜕f (Ω) = f (𝜕Ω). This explains why deg(f − y0 , Ω) is well defined. Invariance of the degree under changes of variable. Assume that ϕ : ℝd → ℝd is a homeomorphism with ϕ(0) = 0 and f ∈ ℳ⋆ (Ω). Then ϕ ∘ f ∘ ϕ−1 ∈ ℳ⋆ (ϕ(Ω)) and deg(f , Ω) = deg(ϕ ∘ f ∘ ϕ−1 , ϕ(Ω)). This property explains why the degree is a topological notion, since it is preserved by conjugacy.

6.3 Computation of the degree 6.3.1 The degree in one dimension Assume that d = 1. If Ω is connected, then Ω = ]a, b[ and the degree is defined by 1 if f (a) < 0 < f (b), { { { deg(f , Ω) = {−1 if f (a) > 0 > f (b), { { {0 otherwise. Exercise 75. Deduce this definition from the axioms of degree theory. When Ω has a finite number of connected components, Ω = ⋃Ni=1 ]ai , bi [, a1 < b1 ≤ a2 < b2 ≤ ⋅ ⋅ ⋅ ≤ aN < bN , the additivity property implies that N

deg(f , Ω) = ∑ deg(f , ]ai , bi [). i=1

Finally, we treat the case of an infinite number of components, Ω = ⋃∞ i=1 Ii , where {Ii } is a family of pairwise disjoint open and bounded intervals. Since f does not vanish on the boundary of Ω, there exists an integer N ≥ 1 such that f does not vanish on the closure of ⋃i>N Ii . Then N

deg(f , Ω) = deg(f , ⋃ Ii ). i=1

6.3 Computation of the degree

| 151

6.3.2 Degree on a disk Assume now that d = 2 and Ω = Ri (Γ), where Γ is a Jordan curve in ℝ2 . In this case the degree can be computed as a winding number. Given f ∈ ℳ⋆ (Ω) we compute the number of revolutions of f (Γ) around the origin. See Figure 6.2. More precisely, we parameterize Γ by α : [0, 1] → ℝ2 with α continuous, α|[0,1[ one-to-one, α(0) = α(1), α([0, 1]) = Γ. In addition we choose the positive (counter-clockwise) orientation1 for the curve Γ. By assumption the parametric curve f ∘ α does not pass through the origin and we can find a continuous argument function θ : [0, 1] → ℝ, f (α(t)) = (cos θ(t), sin θ(t)), |f (α(t))| Then deg(f , Ω) =

t ∈ [0, 1].

1 (θ(1) − θ(0)). 2π

Figure 6.2: deg(f , Ω) = −2.

When α and f are C 1 , this increment of the angle can be expressed by the integral 1

β̇ (s)β1 (s) − β̇ 1 (s)β2 (s) 1 deg(f , Ω) = ds ∫ 2 2π β1 (s)2 + β2 (s)2 0

with βi (s) = fi (α1 (s), α2 (s)). This formula appears in the theory of Kronecker index, an older formulation of degree theory for C 1 functions. 6.3.3 Degree of a linear map Given a linear map L : ℝd → ℝd with non-zero determinant, we observe that L ∈ ℳ⋆ (Ω) for any Ω such that the origin is not in 𝜕Ω. The group GL(ℝd ) has two connected 1 The index of a point p ∈ Ω with respect to α(t) is +1; this index can be understood in the sense of complex analysis.

152 | 6 Appendix on degree theory components, corresponding to linear maps with positive or negative determinant. Every continuous path {Lλ }λ∈[0,1] in GL(ℝd ) defines a homotopy between L0 and L1 . For this reason the degree of a linear map only depends on the sign of the determinant. Indeed, sign(det L) if 0 ∈ Ω, deg(L, Ω) = { 0, if 0 ∈ ̸ cl(Ω). Exercise 76. Consider the linear map D(x1 , . . . , xd−1 , xd ) = (x1 , . . . , xd−1 , −xd ). Use the axioms of degree to prove that deg(D, Ω) = −1, where Ω = {x ∈ ℝd : |xi | < 1, i = 1, . . . , d}. Exercise 77. Deduce the formula for the degree of a linear map from the axioms of degree. The above formula can be extended to affine maps. Given L ∈ GL(ℝd ) and x⋆ ∈ ℝd , the map A(x) = L(x − x⋆ ) belongs to ℳ⋆ (Ω) if x⋆ ∈ ̸ 𝜕Ω. Then sign(det L) if x⋆ ∈ Ω, deg(A, Ω) = { 0 if x⋆ ∈ ̸ cl(Ω). To prove this formula when x⋆ ∈ Ω, it is sufficient to consider a large open and bounded set Ω̃ containing Ω and the segment joining the origin and x⋆ . Then H(x, λ) = L(x − λx⋆ ) is an admissible homotopy and ̃ deg(A, Ω)̃ = deg(L, Ω). By excision, deg(L, Ω)̃ = deg(L, Ω). 6.3.4 Linearization property Assume that f : cl(Ω) → ℝd is C 1 and has a unique zero which is non-degenerate. This means that there exists x⋆ ∈ Ω such that f (x⋆ ) = 0,

det f 󸀠 (x⋆ ) ≠ 0,

f (x) ≠ 0 if x ∈ cl(Ω) \ {x⋆ }.

Then f ∈ ℳ⋆ (Ω) and deg(f , Ω) = deg(f 󸀠 (x⋆ ), Ω) = sign(det f 󸀠 (x⋆ )). Exercise 78. Prove the above formula.

6.4 Fixed point index and some principles | 153

Assume now that f : cl(Ω) → ℝd is C 1 and has a finite number of zeros and all of them are non-degenerate; that is, there exist p1 , . . . , pk ∈ Ω such that f (pi ) = 0,

det f 󸀠 (pi ) ≠ 0,

f (x) ≠ 0 if x ∈ cl(Ω) \ {p1 , . . . , pk }.

Then f ∈ ℳ⋆ (Ω) and k

deg(f , Ω) = ∑ sign(det f 󸀠 (pi )). i=1

This formula is a consequence of the additivity property and it is based on Nagumo’s method for the construction of the degree. Every map f ∈ ℳ⋆ (Ω) can be approximated by maps fn in the above conditions, meaning that fn → f uniformly in cl(Ω), each fn is C 1 in cl(Ω) and it has a finite number of zeros, all of them lying in Ω and nondegenerate. The passage to the limit deg(f , Ω) = lim deg(fn , Ω) n→∞

is valid. Exercise 79. Assume that f ∈ ℳ⋆ (Ω) and L is a linear isomorphism of ℝd . Prove that L ∘ F ∈ ℳ⋆ (Ω) and deg(L ∘ F, Ω) = sign(det L) deg(F, Ω). 6.3.5 The degree of a Cartesian product Assume that d = d1 + d2 and Ω1 ⊂ ℝd1 , Ω2 ⊂ ℝd2 are open and bounded sets. Given f1 ∈ ℳ⋆ (Ω1 ) and f2 ∈ ℳ⋆ (Ω2 ) we define the product map f1 × f2 : Ω1 × Ω2 → ℝd ,

(f1 × f2 )(x, y) = (f1 (x), f2 (y)).

Then f1 × f2 ∈ ℳ⋆ (Ω1 × Ω2 ) and deg(f1 × f2 , Ω1 × Ω2 ) = deg(f1 , Ω1 ) ⋅ deg(f2 , Ω2 ). Exercise 80. Prove this formula.

6.4 Fixed point index and some principles Given a continuous map F : cl(Ω) → ℝd without fixed points on the boundary, F(x) ≠ x

if x ∈ 𝜕Ω,

(6.1)

154 | 6 Appendix on degree theory the fixed point index of F is the degree of id − F; that is, deg(id − F, Ω). This index has a topological character. This means that deg(id − F, Ω) = deg(id − ϕ ∘ F ∘ ϕ−1 , Ω1 ), if ϕ : G → G1 is a homeomorphism between two open subsets of ℝd . Here Ω1 = ϕ(Ω) and cl(Ω) ⊂ G, F(cl(Ω)) ⊂ G1 . Since ϕ(cl(Ω)) is compact, the set Ω1 is bounded. Moreover, from (6.1), we deduce that id − ϕ ∘ F ∘ ϕ−1 belongs to ℳ∗ (Ω1 ). In the rest of this section we always assume that the open and bounded set Ω ⊂ ℝd is also convex. The key to compute the fixed point index of some maps will be the following property of convex domains: Given x0 ∈ Ω and y ∈ cl(Ω), the half-closed segment [x0 , y[ is contained in the domain Ω. Exercise 81. Prove this property. Theorem 19 (Brouwer’s principle). Assume that F : cl(Ω) → ℝd is a continuous map satisfying the condition (6.1) and F(cl(Ω)) ⊂ cl(Ω).

(6.2)

Then deg(id − F, Ω) = 1. Proof. Let us fix a point x0 ∈ Ω and consider the linear homotopy λ(x − F(x)) + (1 − λ)(x − x0 ) = 0, equivalent to x = λF(x) + (1 − λ)x0 . We claim that there are no solutions with x ∈ 𝜕Ω and λ ∈ [0, 1]. For λ = 1 this is precisely the condition (6.1). If λ ∈ [0, 1[ we observe that the point lies on the segment [x0 , y[ with y = F(x). From (6.2) we know that y ∈ cl(Ω) and we can apply the property of convex domains mentioned above. The segment [x0 , y[ is contained in Ω and the point x lies in this segment. In the previous proof the condition (6.2) can be replaced by F(𝜕Ω) ⊂ cl(Ω). Exercise 82. Using the above principle prove Brouwer’s fixed point theorem: a continuous map F : cl(Ω) → cl(Ω) has a fixed point. We are going to present now a related principle involving the iterates of F. To simplify the statement we assume that the domain of F is the whole space ℝd .

6.5 Degree and invariant curves | 155

Theorem 20 (Browder’s principle). Assume that F : ℝd → ℝd is a continuous map and there exists an integer n0 ≥ 1 such that for each n ≥ n0 the iterates F n = F ∘⋅ ⋅ ⋅n times ⋅ ⋅ ⋅∘F satisfy F n (cl(Ω)) ⊂ cl(Ω), F n (x) ≠ x

if x ∈ 𝜕Ω.

(6.3) (6.4)

Then id − F ∈ ℳ∗ (Ω) and deg(id − F, Ω) = 1. Proof. We apply Brouwer’s principle to F n with n ≥ n0 . Then id − F n ∈ ℳ∗ (Ω) and deg(id − F n , Ω) = 1. Since the set of fixed points Fix(F) is contained in Fix(F n0 ), we deduce that id − F ∈ ℳ∗ (Ω). A well-known property of the fixed point index of iterates implies that deg(id − F p , Ω) ≡ deg(id − F, Ω) (mod p) for each prime number p ≥ 2. This congruence and the previous computation lead to ν := deg(id − F, Ω) ∈ {np + 1 : n ∈ ℤ} for each prime number with p ≥ n0 . Let us choose a prime number p > max(n0 , |ν|), then we conclude that ν = 1. A direct consequence of the above result is the following asymptotic fixed point theorem. Every continuous map F : ℝd → ℝd satisfying for n ≥ n0 the conditions (6.3) and (6.4) has a fixed point lying in Ω. It is not sufficient to assume that the conditions (6.3) and (6.4) are satisfied by a sequence of integers nk → +∞. This is illustrated by the following example. Consider the autonomous system of differential equations in the plane ẋ = −x,

ẏ = (1 − y2 )y.

This system has three equilibria, two attractors A± = (0, ±1) and a saddle at the origin. The associated flow {ϕt } is globally defined for the future t ≥ 0. Fix τ > 0 and consider the map F = S ∘ ϕτ with S(x, y) = (x, −y). See Figure 6.3. The condition (6.4) holds for each n ≥ 1 when the domain is Ω = ]− 21 , 21 [ × ] 21 , 32 [. The condition (6.3) holds when n is even. It is clear that the map F has no fixed points in Ω.

6.5 Degree and invariant curves We use the previous properties to give some applications related to the notion of invariant curve in the plane.

156 | 6 Appendix on degree theory

Figure 6.3: F has no fixed points in Ω.

Proposition 16. Assume that G is an open subset of ℝ2 and Γ ⊂ ℝ2 is a Jordan curve with Γ ∪ Ri (Γ) ⊂ G. Let h : G → ℝ2 be a continuous map such that (a) Γ is positively invariant under h, h(Γ) ⊂ Γ; (b) h has no fixed points on Γ, h(x) ≠ x if x ∈ Γ. Then deg(id − h, Ri (Γ)) = 1. Proof. The disk Γ ∪ Ri (Γ) is transformed onto the unit disk 𝔻 via some ϕ ∈ ℋ(ℝ2 ). The map ĥ = ϕ ∘ h ∘ ϕ−1 is defined on the open set Ĝ = ϕ(G) and 𝔻 is contained in G.̂ Moreover, the unit circle is positively invariant under ĥ and Fix(h)̂ ∩ 𝜕𝔻 = 0. We can apply the remark after the proof of Theorem 19 to deduce that deg(id − h,̂ int(𝔻)) = 1. The topological character of the degree implies that deg(id − h, Ri (Γ)) = deg(id − h,̂ int(𝔻)) = 1. Remark 1. The previous result admits an obvious extension to arbitrary dimension. Assume that S ⊂ ℝd is a tame sphere; i. e. there exists ϕ ∈ ℋ(ℝd ) such that ϕ(S) = 𝕊d−1 . Let Ω be the bounded component of ℝd \ S and h : Ω ∪ S → ℝd a continuous map with h(S) ⊂ S and Fix(h) ∩ S = 0. Then deg(id − h, Ω) = 1. Proposition 17. Assume that G is an open subset of ℝ2 and let X : G → ℝ2 be a continuous vector field such that there is uniqueness for the initial value problem associated to the system of differential equations ẋ = X(x). Assume that Γ is a closed orbit of this system with Γ ∪ Ri (Γ) ⊂ G. Then deg(X, Ri (Γ)) = 1. Proof. Take a point p ∈ Γ. By assumption x(t, p) is a non-constant periodic solution and Γ = {x(t, p) : t ∈ ℝ}.

6.6 Bibliographical remarks | 157

For each t ∈ ℝ let us consider the continuous map Pt (p) = x(t, p). It is well defined on Γ ∪ Ri (Γ) and Γ is invariant under Pt . Assume that T > 0 is the minimal period of x(t, p), then Pt has no fixed points on Γ if t ∈ ]0, T[. For t ∈ [0, T], the map Pt is well defined in some neighborhood G of Γ ∪ Ri (Γ). It follows from Proposition 16 that deg(id − Pt , Ri (Γ)) = 1

if 0 < t < T.

Let us fix τ ∈ ]0, T[ and consider the map H : (Ri (Γ) ∪ Γ) × [0, 1] → ℝ2 ,

1 (p − Pt (p)), t ∈ ]0, τ[ H(p, t) = { t −X(p), t = 0.

The identity t

1 H(p, t) = − ∫ X(x(s, p))ds, t

if t ≠ 0,

0

is useful to prove the continuity of H. Since H does not vanish on Γ × [0, τ] we can invoke the homotopy property to deduce that deg(−X, Ri (Γ)) = deg(id − Pτ , Ri (Γ)) = 1. Finally, we observe that ℋ(p, λ) = R[λπ]X(p) defines an admissible homotopy between X and −X. Recall that cos θ R[θ] = [ sin θ

− sin θ ]. cos θ

Therefore deg(−X, Ri (Γ)) = deg(X, Ri (Γ)). Alternatively we can apply Exercise 79.

6.6 Bibliographical remarks As regards Section 6.1, it is traditional to define the degree for a triplet (f , Ω, p), where p is a point in ℝd . In our notation d[f , Ω, p] = deg(f − p, Ω). Following the book by Granas and Dugundji [50], we have preferred to define the degree for couples (f , Ω), assuming implicitly that p = 0. The axiomatics of degree is discussed in detail in [50]. For different approaches to the construction of the degree we refer to the books [42, 64, 79, 110]. As regards Section 6.2, a very clear discussion on the consequences of the product theorem by Leray can be found in [110].

158 | 6 Appendix on degree theory As regards Section 6.3, the definition of the degree on a disk as a winding number is based on the existence of a continuous argument for a path in 𝕊1 . This is a particular case of the lifting property of covering spaces and we refer to [84] for more details. The degree for C 1 maps was essentially defined by Kronecker in terms of an integral and it was a well-known notion around 1900. For instance, a complete chapter (Chapter 7, Volume 2) of the Analysis Course by Picard [105] was devoted to the Kronecker index. As regards Section 6.4, for a proof of the topological character of the fixed point index we refer to Theorem 26.4 in [64]. The term Browder’s principle is coined in the book [64]. The proof of Proposition 20 is taken from this book. The property of the iterates of the fixed point index can be proved using techniques of differential topology as in [64] or using algebraic topology. This second approach can be found in [50]. See the mod p theorem in Chapter 5 of this book. Another reference on this topic is [60]. The asymptotic fixed point theorem was originally presented by Browder [20] in the more general setting of the Leray–Schauder degree. We refer to the final comments of Chapter V of [50] and to the recent paper [124] for more information on the origin of the so-called asymptotic fixed point theory. The property of convex sets is also used in Chapter 4, Section 1.11 of the book [110]. As regards Section 6.5, the fixed point index inside an invariant curve (Proposition 16) was first studied by Lifschitz in his paper [78]. As explained in the paper, the original motivation came from the study of the Poincaré–Birkhoff fixed point theorem. As we have seen in Chapter 5, this proposition also plays a role in the proof of the arc translation lemma. The result by Lifschitz is also discussed by Lefschetz in his book on differential equations [72]. Probably, for a more expert audience, he could have mentioned that it is a very direct consequence of his famous fixed point theorem. In Lefschetz’s opinion, the main merit of this index computation is the obtention of Proposition 17 as a limiting case. As mentioned in [72], Proposition 17 already appeared in Poincaré’s work.

7 Solutions to the exercises 7.1 Chapter 1 Exercise 1 Consider the initial value problem ü + u1/3 = 0,

u(0) = u0 ,

̇ u(0) = v0 .

We prove that locally there is a unique solution for each initial condition (u0 , v0 ) ∈ ℝ2 . From this it is easy to deduce that the uniqueness is global. If u0 ≠ 0 the function f (u) = u1/3 is C 1 in a neighborhood of u0 and standard uniqueness results are applicable. Next assume that u0 = 0 but v0 ≠ 0 (say v0 < 0). Then u(t) is also a solution of

u̇ = g(u), u(0) = 0 where g(u) = −√2E − 32 u4/3 , E = 21 v02 > 0. The function g(u) is C 1 in a neighborhood of u = 0 and we apply again standard results, now to the first order problem. Finally, assume that u0 = v0 = 0. From the conservation of energy, 1 3 ̇ 2 + u(t)4/3 = 0 u(t) 2 4 everywhere. Then u(t) ≡ 0. For the equation ü − u1/3 = 0, observe that the trivial solution and u(t) = 6−3/2 t 3 share the initial conditions at t = 0. Exercise 2 Assume d = 1 and X(t, x) = 1 + x2 . Then x(t, p) = tan(t + arctan(p)), ω(p) = π2 − arctan(p) and 𝒟 = 0 if T ≥ π. Assume d = 2 and identify ℂ with ℝ2 , the system ż = 1 + z 2 can be integrated as before. By direct computation it is possible to check that 𝒟 = ℂ \ ℝ if T ≥ π. For d = 1 and t fixed, p 󳨃→ x(t, p) is increasing and therefore ω(p) is monotone non-increasing. This implies that 𝒟 is an interval. For d = 2 let 𝒟̂ be a connected component of 𝒟. We prove that if Γ is a Jordan curve contained in 𝒟̂ , then Ri (Γ), the bounded component of ℝ2 \ Γ, is also contained in 𝒟̂ . Define C = {(x(t, p), t) ∈ ℝ2 × [0, T] : p ∈ Γ, t ∈ [0, T]}. This is a topological cylinder splitting (ℝ2 × [0, T]) \ C in two connected components ℬ and 𝒰 , one bounded and another unbounded. By uniqueness, the graph of a solution x(t, p) with p ∈ Ri (Γ) cannot cross C and so it remains in ℬ if t < min{T, ω(p)}. In particular the solution cannot blow up and therefore ω(p) > T. We have proved that Ri (Γ) ⊂ 𝒟. The closure cl(Ri (Γ)) is a connected subset of 𝒟 with cl(Ri (Γ))∩ 𝒟̂ ≠ 0. Hence cl(Ri (Γ)) ⊂ 𝒟̂ . https://doi.org/10.1515/9783110551167-007

160 | 7 Solutions to the exercises Exercise 3 From the formula of variation of constants we deduce that the Poincaré map is affine. Namely, P(p) = Φ(T)p + η T

where Φ(t) is the matrix solution of Ẋ = A(t)X, X(0) = Id and η = Φ(T) ∫0 Φ−1 (s)b(s)ds. The map P has a fixed point if and only if the linear system Mp = η is compatible, where M = Id − Φ(T). Given σ ∈ Ker(M ⋆ ), ⟨P(p), σ⟩ = ⟨Φ(T)p, σ⟩ + ⟨η, σ⟩ = ⟨p, Φ(T)⋆ σ⟩ + ⟨η, σ⟩ = ⟨p, σ⟩ + ⟨η, σ⟩. In consequence, ⟨P n (p), σ⟩ = ⟨p, σ⟩ + n⟨η, σ⟩. If the equation has a bounded solution x(t, p), the sequence {P n (p)}n≥0 must be bounded. Then ⟨η, σ⟩ = 0 and η is orthogonal to Ker(M ⋆ ). The system Mp = η is compatible. Exercise 4 Consider ẋ = x + sin t. Then ϕt (p) = (p + 21 )et − 21 (cos t + sin t) and ϕ2π ∘ ϕπ ≠ ϕπ ∘ ϕ2π . Exercise 5 Let Ψ : [0, 1] → GL⋆ (ℝd ) be a smooth path such that Ψ = id on [0, δ] and Ψ = L on −1 ̇ [1 − δ, 1] for some small δ. When A(t) = Ψ(t)Ψ(t) the solution is ϕt (p) = Ψ(t)p. Remark: note that sometimes L has not a real logarithm and the matrix A(t) cannot be constant. Exercise 6 Consider the three diffeotopies ℋi = ℋi (t, p), i = 1, 2, 3 constructed in Lemma 1. If det P 󸀠 = 1 everywhere, the first two diffeotopies satisfy det(

𝜕ℋi (t, p)) ≡ 1, 𝜕p

i = 1, 2.

Also the third, ℋ3 , can be chosen with this property. This follows from the hint to the exercise. The proof of Theorem 1 will produce an equation ẋ = X(t, x) with 𝜕ϕ det( 𝜕p (t, p)) ≡ 1. From the Jacobi–Liouville formula, t

1 = e∫0 divx X(s,ϕ(s,p))ds .

7.2 Chapter 2

| 161

Hence, for each t, p and h ≠ 0, t+h

1 ∫ divx X(s, ϕ(s, p))ds = 0 h t

and, letting h → 0, divx X(t, ϕ(t, p)) = 0. Since ϕt (ℝd ) = ℝd we conclude that divx X = 0. Exercise 7 Introduce the notation z = (x, y), Pϵ (z) = P(ϵ, z). Then Pϵ is a diffeomorphism with inverse Pϵ−1 (x1 , y1 ) = (x1 − ϵy1 + ϵ2 sin x1 , y1 − ϵ sin x1 ). Moreover, det Pϵ󸀠 (z) ≡ 1. Let χ(t) be a function as in the proof of Theorem 1. The periodic system ̇ ż = Z(t, z) = χ(t)

𝜕P −1 (z)) (χ(t), Pχ(t) 𝜕ϵ

has the periodic flow ϕ(t, p) = Pχ(t) (p). By the same argument of Exercise 6, divz Z ≡ 0. 𝜕H 𝜕H Then there exists H = H(t, z) with 𝜕z = −Z2 , 𝜕z = Z1 . 1

2

7.2 Chapter 2 Exercise 8 Let u(t, u0 ) be the solution with u(0) = u0 . We say that there is extinction if ω(u0 ) = +∞ and limt→+∞ u(t, u0 ) = 0 for every u0 ∈ ℝ+ . Let us observe that every constant u ≡ c ≥ 3 is an upper solution while u ≡ 0 is a solution. Then every solution will satisfy 0 ≤ u(t) ≤ max{u(0), 3} if t ∈ [0, ω(u0 )[. In consequence, all solutions are defined and bounded in [0, ∞[. We know from Theorem 2 that all of them are asymptotically T-periodic. Assume that there is no extinction. For some u0 > 0 we have lim supt→+∞ u(t, u0 ) > 0. Let φ(t) be the T-periodic solution such that u(t, u0 ) − φ(t) → 0 as t → +∞. Then φ(t) ≥ 0 is a non-trivial T-periodic solution and, by uniqueness, φ(t) > 0 everywhere. Exercise 9 When x1 = x0 the point x0 is fixed and we take q = x0 . Assume now that x1 ≠ x0 , say x1 > x0 . Then x2 = P(x1 ) > P(x0 ) = x1 and, by induction, we conclude that {xn } is an increasing sequence contained in K. Let q be the limit of this sequence. Then, letting n → +∞ in xn+1 = P(xn ) we conclude that q is a fixed point. Theorem 2 is a direct consequence if we employ Section 1.2 of Chapter 1.

162 | 7 Solutions to the exercises Exercise 10 The function φ(t) = sin( π2 sin t) is a global solution of this equation. Also the constants x± = ±1. There is not uniqueness for this equation since φ(t) and x+ (t) coincide at the instants tn = π2 + 2nπ. This fact allows us to define new solutions by gluing φ(t) and x+ (t). As an example consider the sequence of interval In = [ π2 + 2π(n2 − 1), π2 + 2πn2 ] for each n ≥ 1. They are pairwise disjoint and we can define φ(t), x(t) = { 1,

t ∈ ⋃n≥1 In , otherwise.

Then x(t) is not asymptotically periodic. Exercise 11 Assume that x(t) is a solution with |x(t)| ≤ C if t ∈ [τ, +∞[. The translate xn (t) = x(t + nT) is a solution defined on [τ − nT, +∞[ with |xn (t)| ≤ C on this interval. Let us select a convergent sub-sequence of {xn (0)}n≥0 , say xσ(n) (0) → ξ . Let y(t) be the solution with initial condition y(0) = ξ and maximal interval ]α, ω[. Given a compact interval J ⊂ ]α, ω[ we observe that xσ(n) (t) is well defined on J if n is large enough. By continuous dependence xσ(n) (t) → y(t) if t ∈ J. In consequence |y(t)| ≤ C if t ∈ ]α, ω[. The general theory of continuation of solutions implies that ]α, ω[ = ℝ. Once we have a solution y(t) defined globally and bounded we consider φN (t) = sup{y(t + nT) : |n| ≤ N}. This is a solution satisfying |φN (t)| ≤ C, φN+1 (t) ≥ φN (t), φN+1 (t) ≥ φN (t + T) ≥ φN−1 (t). It is easy to prove that φ(t) = limN→+∞ φN (t) is a T-periodic solution. Exercise 12 The general theory of maximal solutions implies that limt↑ω |x(t)| = ∞ if ω < +∞. Assume that ω = +∞ and x(0) ≤ x(T). Then, by uniqueness, x(t) ≤ x(t + T) if t ≥ 0. Iterating this inequality, x(t) ≤ x(t + T) ≤ x(t + 2T) ≤ ⋅ ⋅ ⋅ ≤ x(t + nT) ≤ ⋅ ⋅ ⋅ if t ∈ [0, T]. Therefore inf[0,∞[ x(t) = min[0,T] x(t) and lim inft↑ω x(t) > −∞. The case x(0) > x(T) is treated similarly. Now, lim inft↑ω x(t) < +∞. Exercise 13 The function f can be split as f = φ + r with φ : ℝ → ℝ continuous and p-periodic and r : [0, ∞[ → ℝ vanishing at infinity, limt→+∞ r(t) = 0. We will prove that both limits ̂ exist. Moreover, φ(λ) ̂ ̂ ̂ φ(λ) and r(λ) = 0 if λ ∈ ̸ ωℤ and r(λ) = 0 for all λ. From now on λ ∈ ℝ is a fixed number. A direct computation shows that T

Np

0

0

1 1 ∫ e−iλt φ(t)dt → 0 ∫ e−iλt φ(t)dt − T Np

7.2 Chapter 2

| 163

as T → +∞, with N the integer part of Tp . By periodicity, Np

p

0

0

1 N−1 −iλkp 1 ) ∫ e−iλt φ(t)dt. (∑ e ∫ e−iλt φ(t)dt = Np Np k=0 −iλkp = O(1) Summing the geometric sequence with ratio e−iλp , we observe that ∑N−1 k=0 e N−1 −iλkp ̂ = N if λ ∈ ωℤ. The conclusion for φ(λ) follows. as N → +∞ if λ ∈ ̸ ωℤ and ∑k=0 e Let us now work with r(t). Given ϵ > 0 we find τ > 0 such that |r(t)| ≤ ϵ2 if t ≥ τ. Then, for T large enough,

󵄨󵄨 󵄨󵄨 T 󵄨 󵄨󵄨 1 󵄨󵄨 ∫ e−iλt r(t)dt 󵄨󵄨󵄨 ≤ ‖r‖∞ τ + ϵ T − τ < ϵ. 󵄨󵄨 󵄨󵄨 T T 2 T 󵄨󵄨 󵄨󵄨 0 Exercise 14 For d = 2 we define Φ(x) = y with π y(t) = R[ t]x(t). T Next we prove that a similar isomorphism cannot be constructed for d = 1. First we observe that every non-trivial anti-periodic function must have at least one zero in [0, T]. This is a consequence of the continuity. In consequence we have the inequality ‖x‖1 < T‖x‖∞

if x ∈ 𝒜 \ {0}.

Assume by contradiction that Φ exists and consider x = Φ−1 (1), where 1 ∈ 𝒫 denotes the constant function. The isometry properties of Φ imply that ‖x‖∞ = ‖1‖∞ = 1, ‖x‖1 = ‖1‖1 = T. This is incompatible with the previous inequality. Exercise 15 Consider the scalar equation ẏ = Y(t, y),

with Y(t, y) := Xd (t, x1 (t), . . . , xd−1 (t), y).

This is a periodic equation with d = 1 and Massera’s convergence theorem applies. The solution y(t) = xd (t) is bounded and therefore we know that there exists a T-periodic solution φ(t) such that, for each t ∈ ℝ, xd (t+nT)−φ(t) → 0 as n → +∞. The periodicity of (2.1) implies that for each n ≥ 0 the vector-valued function (x1 (t + nT), . . . , xd−1 (t + nT), xd (t + nT)) = (x1 (t), . . . , xd−1 (t), xd (t + nT)) is a solution. By a passage to the limit (n → +∞) we conclude that also (x1 (t), . . . , xd−1 (t), φ(t)) is a solution of (2.1).

164 | 7 Solutions to the exercises Exercise 16 Consider the autonomous system ẋ1 = cos x2 ,

ẋ2 = − sin2 x2 .

All solutions are defined in ]−∞, +∞[. There are two types of orbits: the horizontal lines x2 = nπ with n an integer and the curves of equation x1 − sin1 x = C with C ∈ ℝ 2 with C ∈ ℝ. See Figure 7.1.

Figure 7.1: Phase portrait.

It is clear that all solutions go to infinity as |t| → ∞. Let us prove that the limit limt→+∞ ‖x(t, p)‖ = ∞ is not uniform on the ball ‖p‖ < π. A rescaling of the system will reduce this ball to the unit ball. Define pn = (0, π − n1 ). We observe that |x2 (t, pn )| < π if t ∈ ℝ and x1 (t, pn ) → −∞ as t → +∞. Also, there exists a unique τn > 0 such that x1 (τn , pn ) = 0. By continuous dependence the orbit passing through pn must remain close to y = π for a large time. Then τn → +∞ and this shows that the limit limt→+∞ ‖x(t, pn )‖ = ∞ is not uniform in n. Exercise 17 Let u(t) = (u1 (t), u2 (t)) be a solution of the prey–predator system with u1 (t0 ) > 0, u2 (t0 ) > 0. We claim that it is well defined [t0 , ∞[ and there exists a number M > 0, depending on u, such that 0 < ui (t) ≤ M

if t ∈ [t0 , ∞[, i = 1, 2.

By uniqueness of the initial value problem ui (t) > 0 if t ∈ [t0 , ω[. From the first equation we obtain the differential inequality u̇ 1 ≤ u1 (a − bu1 ) with a = maxt∈ℝ a(t), b = mint∈ℝ b(t) > 0. Then u1 (t) ≤ U(t) if t ∈ [t0 , ω[, where U(t) is the solution of the logistic equation U̇ = U(a − bU) with U(t0 ) = u1 (t0 ). We define M1 := supt≥t0 U(t) < ∞. Similarly, u̇ 2 ≤ u2 (d + eM1 − f u2 ) and we get an analogous estimate u2 (t) ≤ M2 < ∞ if t ∈ [t0 , ω[. This implies that ω = +∞ and we can take M = {M1 , M2 }. After these preliminary remarks we are ready to prove the result by a contradiction argument, assuming that some solution satisfies lim sup min{u1 (t), u2 (t)} > 0. t→+∞

7.2 Chapter 2

| 165

We transport the system from Ω = ]0, ∞[ × ]0, ∞[ to ℝ2 via ui = exi . For the system in x we know that all the solutions are defined in the future and the solution x(t) = (ln u1 (t), ln u2 (t)) satisfies lim inft→+∞ ‖x(t)‖ < ∞. From Theorem 4 we deduce that there exists a 1-periodic solution of the original system lying in Ω. Exercise 18 ̇ is bounded on the interval [0, ∞[. This The first step will be to prove that z(t) = x(t) function is a solution of the first order linear equation ż + cz = p(t) with p(t) = β sin t − sin x(t). Then, for some k ∈ ℝ, z(t) = e

−ct

t

k + ∫ e−c(t−s) p(s)ds. −∞

Since ‖p‖∞ ≤ β + 1 we deduce that |z(t)| ≤ k + c1 (β + 1) if t ≥ 0. The couple (Θ(t), z(t)) ̇ is a solution of the system (in ℝ2 ) with Θ(t) = x(t) − pq t, z(t) = x(t) η̇ = ξ −

p , q

ξ ̇ = −cξ − sin(η +

p t) + β sin t. q

This system has period T = 2πq and all solutions are globally defined. Since a solution is bounded in the future, we can apply Massera’s theorem to deduce the existence of a 2πq-periodic solution (η∗ (t), ξ∗ (t)). Then x∗ (t) = η∗ (t) + pq t is a solution of the forced pendulum equation satisfying x∗ (t + 2πq) = x∗ (t) + 2πp. Exercise 19 First example at the end of Section 1.2. Exercise 20 Fix T = 2π and consider the equation ẍ + ω2 x + (ω2 − ẋ 2 − ω2 x2 ) sin t = 0 with ω = The solution x(t) = sin ωt has minimal period σ.

2π . σ

Exercise 21 For simplicity assume that T = 1. Given a real number x, let {x} denote the fractional part; that is, x = [x] + {x}, where [x] is the integer part. Recall that the set Dσ = {{nσ} : n ∈ ℤ} is dense in [0, 1] if σ is irrational. The σ-periodicity of φ(t) ̇ 0 ) = φ(t ̇ 0 + nσ) = X(t0 + nσ, φ(t0 )). The 1-periodicity of implies that X(t0 , φ(t0 )) = φ(t X(⋅, φ(t0 )) implies that X(t0 + nσ, φ(t0 )) = X(t0 + {nσ}, φ(t0 )). In consequence X(⋅, φ(t0 )) is constant on t0 + Dσ and, by continuity, X(⋅, φ(t0 )) is also constant on [t0 , t0 + 1].

166 | 7 Solutions to the exercises

7.3 Chapter 3 Exercise 22 Any set Ω ⊂ ℝ2 that is open and simply connected is the image of some embedding. In [93] it is proved that Ω is homeomorphic to the plane. Alternatively, we can employ Riemann’s theorem on conformal mappings, to map Ω onto the open unit disk via a x holomorphic diffeomorphism R. We define h = R−1 ∘ h1 where h1 (x) = 1+‖x‖ . Exercise 23 h(Re (Γ)) = h(ℝ2 \ (Ri (Γ) ∪ Γ)) = h(ℝ2 ) \ h(Ri (Γ) ∪ Γ) = Ω \ (Ri (h(Γ)) ∪ h(Γ)) = Ω ∩ Re (Γ). Exercise 24 It is not restrictive to assume that G is bounded. We take p0 ∈ G and q0 = h1 (p0 ) = h2 (p0 ). Then deg(h1 − q0 , G) = deg(h2 − q0 , G). Exercise 25 This is a consequence of Leray’s product theorem. Note that f = h − q is one-to-one and the degree must be ±1 because 0 ∈ f (𝒰 ). Exercise 26 We employ the formula for the degree of a composition with x0 = 0, f = h1 , g = h2 and Ω an open ball containing the origin. Then h1 (Ω) is a neighborhood of y0 = h1 (0). Since h1 and h2 preserve the orientation, deg(h1 − y0 , Ω) = deg(h2 − z0 , h1 (Ω)) = 1. Then also deg(h2 ∘ h1 − z0 , Ω) = 1. Exercise 27 In view of Exercise 26, we can restrict ourselves to the case h ∈ ̸ ℰ∗ (ℝ2 ). Then we can use the formula for the degree of a composition to prove that h1 = h∘S and h2 = S ∘h belong to ℰ∗ (ℝ2 ). Here S(x1 , x2 ) = (x1 , −x2 ). Since h2 = h1 ∘ h2 , we apply again Exercise 26. Exercise 28 Consider an increasing homeomorphism f : [0, ∞[ → [0, ∞[ with f (r) > r if 0 < r < 1 and f (r) < r if r > 1. Define the sequence ρn+1 = f (ρn ), ρ0 = 21 . This sequence converges to 1 and we can construct a continuous function g : [0, ∞[ → [0, ∞[ such that g(1) = 0 and ∑n≥0 g(ρn ) = ∞. Define h in polar coordinates, θ1 = θ + g(r),

r1 = f (r).

For each p ≠ 0, Lω (p) ⊂ 𝕊1 ⊂ Fix(h) = 𝕊1 ∪ {0} and so h has trivial dynamics. For ‖p‖ = 21 , Lω (p) = 𝕊1 . Note that θn+1 − θn = g(ρn ) → 0 and θn = θ0 + ∑k x⋆ . Then h(x) < x⋆ and x⋆ < h2 (x). We take D = [a, b] with x⋆ < a and such that, for some x ∈ D. Also h2 (x) belongs to D. By construction D ∩ h(D) = 0 and D ∩ h2 (D) ≠ 0. Exercise 30 The proof of Proposition 3 is easily adapted. The contraction h(x) = 21 x is locally free in any number of dimensions. Exercise 31 By assumption the set Lω (p) contains the point p which is not fixed. Hence Lω (p) ⊄ Fix(h) and h has not trivial dynamics. From Theorem 6 we deduce that there exists a Jordan curve Γ ⊂ ℝ2 \ Fix(h) with deg(id − h, Ri (Γ)) = 1. The property of stability of the degree under small perturbations implies the existence of a number ϵ > 0 such that ̂ deg(id − h,̂ Ri (Γ)) = 1 if ‖h(p) − h(p)‖ < ϵ for each p ∈ Γ. Therefore ĥ has a fixed point lying in Ri (Γ). Exercise 32 Consider the segment α joining the points p = (−1, 0) and q = (1, 0) and the sub-arc ]}. Let h be a homeomorphism of the of the unit circle β = {(cos θ, − sin θ) : θ ∈ [0, 3π 2 plane sending α onto β. Then h(α \ {p}) ∩ (α \ {p}) = 0 and h(α \ {q}) ∩ (α \ {q}) = {p}. Exercise 33 Define (2x, −y) if x ≤ 0, h(x, y) = { 1 ( 2 x, −y) if x ≥ 0. This map is a homeomorphism of the plane having a unique fixed point at the origin. The arc α = {(cos θ, sin θ) : θ ∈ [− π2 , π2 ]} is a translation arc with hn (α) ∩ α = {(0, 1), (0, −1)} for each n ≥ 1. We prove that deg(id − h, Ri (Γ)) = 0 for each Jordan curve Γ ⊂ ℝ2 \ {0}. To compute this degree we employ the property of invariance under homotopies. Define (−x + λ, 2y) if x ≤ 0, Hλ (x, y) = { 1 ( 2 x + λ, 2y) if x ≥ 0.

168 | 7 Solutions to the exercises The map Hλ : ℝ2 ×[0, 1] → ℝ2 is continuous with H0 = id−h. Moreover, Hλ (x, y) ≠ (0, 0) if λ > 0. Then deg(id − h, Ri (Γ)) = deg(H1 , Ri (Γ)). Since H1 never vanishes this degree is zero. Exercise 34 Otherwise the degree of id − h is 1 on some region and there exists a fixed point. Exercise 35 S = 𝕊1 ∪ ([0, 1] × {0}). Any homeomorphism ϕ under the conditions of the exercise should send the unit disk onto the unit disk, but the segments 𝔸 and [0, 1] × {0} are in different components of the complement. Exercise 36 Define B1 = {x = (x1 , x2 , x3 ) ∈ ℝ3 : 21 ≤ ‖x‖ ≤ 1, x3 ≥ − 41 } and B2 = {x = (x1 , x2 , x3 ) ∈ ℝ3 : 21 ≤ ‖x‖ ≤ 1, x3 ≤ 41 }. Then B1 and B2 are homeomorphic to 𝔹 and B1 ∩ B2 is homeomorphic to the solid torus 𝕊1 × 𝔻. Exercise 37 Let us select a number τ > 0 which is exactly the period of some closed orbit around C+ . Then all the points lying in this orbit are fixed under h1 and Fix(h1 ) is infinite. Exercise 38 It is sufficient to split ℝd with d ≥ 4 as ℝ3 × ℝd−3 and define h = h3 × id. Exercise 39 If K = 𝕊1 ∪ ([0, 1] × {0}), the point (1, 0) is always fixed. Exercise 40 Define Kn = hn (K), then K∞ = ⋂∞ n=0 Kn is an invariant continuum contained in K. ∞ 2 2 Moreover, ℝ \ K∞ = ⋃n=0 (ℝ \ Kn ) is the union of a family of arcwise connected sets ℝ2 \ Kn having a non-empty intersection. This implies that ℝ2 \ K∞ is also arcwise connected. Exercise 41 The set G = ℝ3 \{0} is simply connected. This implies that any covering map ℘ : X̃ → G is a homeomorphism. Since G is not homeomorphic to ℝ3 , X̃ cannot be ℝ3 .

7.4 Chapter 4

| 169

Exercise 42 Let U ⊂ 𝕊2 \ K be a component and let γ ⊂ U be an arbitrary Jordan curve. The connected set K is contained inside one of the two components of 𝕊2 \ γ. Then γ can be deformed homotopically to a point through the other component, which is contained in U.

7.4 Chapter 4 Exercise 43 ẋ = −x, x(t) = e−t . Exercise 44 Assume by contradiction that φ(t) is stable. Then ϵn = ‖φ − φn ‖∞ → 0. Since φ is not T constant, we can select τ > 0 with φ(τ) ≠ φ(0). The condition Tn ∈ ̸ ℚ implies that the set {qT + pTn : q ∈ ℕ, p ∈ ℤ} is dense in ℝ. Then we can find sequences qn ∈ ℕ, pn ∈ ℤ such that qn T + pn Tn → τ as n → ∞. From the definition of τ, 0 < ‖φ(τ) − φ(0)‖ ≤ Δ1 (n) + Δ2 (n) + Δ3 (n) where Δ1 (n) := ‖φ(τ) − φ(qn T + pn Tn )‖, Δ2 (n) := ‖φ(qn T + pn Tn ) − φn (qn T + pn Tn )‖, Δ3 (n) := ‖φn (qn T + pn Tn ) − φ(0)‖ = ‖φn (qn T) − φ(qn T)‖. We observe that Δ1 (n) tends to zero because φ is continuous at τ and Δi (n) ≤ ϵn for i = 2, 3. The desired contradiction is obtained by letting n → ∞. Exercise 45 First we observe that if u(t) is a solution, then also uλ (t) = λu(λt) is a solution for each λ > 0. From the conservation of energy, 1 1 1 1 ̇ 2 + u(t)4 = u(0) ̇ 2 + u(0)4 . u(t) 2 4 2 4 It is not hard to prove that all the solutions are periodic. Note that if u(t) is a nontrivial solution with minimal period T > 0, then Tλ is the minimal period of uλ (t). After transforming the equation in a first order system, x = (u, u)̇ ∗ , we observe that the conservation of energy implies that x ≡ 0 is stable. Let us fix a non-trivial solution ̇ ∗ with minimal period T. We take a sequence of positive numbers λn φ(t) = (u(t), u(t)) ̇ n t)) is a sequence of such that λn → 1 and λn ∈ ℝ \ ℚ. Then φn (t) = (λn u(λn t), λn2 u(λ T periodic solutions with periods Tn = λ and φn (0) → φ(0). We can apply Exercise 44 n to conclude that φ(t) is unstable.

170 | 7 Solutions to the exercises Exercise 46 Consider the system defined in the closed unit disk that can be expressed in polar coordinates by the equations θ̇ = 1, r ̇ = f (r) with f ∈ C ∞ ([0, 1]) and f (0) = f (1) = 0. Assume also that 0 is an accumulation point of the compact set K = {r ∈ [0, 1] : f (r) = 0}. That is, there exists a sequence rn ∈ K with rn > 0 and rn → 0. Under these conditions the origin p∗ = 0 is an isolated and stable equilibrium. The closed orbits are circles with radius r ∈ K. Given f1 and f2 under the previous conditions, the equivalence of X1 and X2 in the closed unit disk implies that closed orbits go onto closed orbits and so the sets K1 and K2 must be homeomorphic. This construction is for the closed disk but it is not hard to adapt it to open domains. Exercise 47 Assume by contradiction that q ≠ p∗ is also a point in the limit set Lω (p). Then we fix ϵ > 0 with ϵ < 21 ‖q − p∗ ‖. The stability of p∗ implies the existence of a number δ > 0 such that if ‖P − p∗ ‖ < δ then ω(P) = +∞ and ‖x(t, P) − p∗ ‖ < ϵ for each t ≥ 0. From the definition of the limit set we know that there exists a sequence τn → ∞ with x(τn , p) → p∗ . Let us select n large enough so that ‖x(τn , p) − p∗ ‖ < δ. Then x(t, p) will remain for t ≥ τn in the open ball of radius ϵ and centered at p∗ . Then the limit set must be contained in the closed ball; this is a contradiction because q is outside this ball. Exercise 48 Consider the splitting ℝd = ℝ3 × ℝd−3 with points z = (x, y), x ∈ ℝ3 , y ∈ ℝd−3 and define the vector field Z : ℝd → ℝd , Z(x, y) = (X(x), −y). Let p∗ ∈ ℝ3 be the equilibrium of X, then z∗ = (p∗ , 0) is an isolated and stable equilibrium of the enlarged system. Given small neighborhoods 𝒰 ⊂ ℝ3 of p∗ and 𝒱 ⊂ ℝd−3 of the origin, deg(Z, 𝒰 × 𝒱 ) = deg(−id, 𝒱 ) deg(X, 𝒰 ) = (−1)d−3 k. Since Z = X × (−id), this is a consequence of the degree formula for the Cartesian product of maps. Exercise 49 By a contradiction argument assume that X̂ ∗ exists and let x(t) be a solution of ẋ = X̂ ∗ (x) with x(0) ∈ D. Then x(t) ∈ P for each t ≥ 0 and the corresponding ω-limit set must remain in the compact set P. Poincaré–Bendixson’s theory is applicable in this setting and it would imply that this limit set is a closed orbit Γ ⊂ P. Here we are using the fact that the vector field has no equilibria. Since we are in the plane and P is simply connected, the vector field should have an equilibrium lying in Ri (Γ) and this is against the condition (iii).

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| 171

Figure 7.2: An equilibrium with zero index.

Exercise 50 The phase portrait associated to X is given by Figure 7.2. We only insert plugs 𝒫n+ . A key observation is that the vector field X is positively homogeneous, X(λx) = λX(x) if λ > 0. To compute the index we observe that the formula Xϵ (x) = X(x) + ϵ(0, 0, 1),

ϵ ∈ [0, 1],

defines a homotopy on any bounded and open set Ω ⊂ ℝ3 with 0 ∈ ̸ 𝜕Ω. Since Xϵ has no zeros if ϵ > 0, deg(X, Ω) = deg(X1 , Ω) = 0. Alternatively we could express X as the Cartesian product of three one dimensional maps. Exercise 51 Fix ρ ∈ ]r, R[ and define X̂ ∗ ((λr + (1 − λ)ρ)ξ1 , (λr + (1 − λ)ρ)ξ2 , x3 ) = λY(rξ1 , rξ2 , x3 ) + (1 − λ)(0, 0, 1) if ξ12 + ξ22 = 1, x3 ∈ [0, h], λ ∈ [0, 1]. Also, X̂ ∗ = X∗ if ρ2 ≤ x12 + x22 ≤ R2 or x3 ∈ [−ϵ, 0] ∪ [h, h + ϵ]. Exercise 52 The spectrum of A is denoted by σ(A) while r(A) denotes the spectral radius. Since p∗ = 0 is isolated it is implicitly assumed that 1 ∈ ̸ σ(A). There are three possible configurations: – σ(A) ⊂ ]1, +∞[; – σ(A) ∩ ]1, +∞[ = 0 and r(A) > 1; – σ(A) = {−1} and A ≠ −I. Exercise 53 Assume that X : U ⊂ ℝ2 → ℝ2 is a Lipschitz-continuous vector field defined on an open set U. Moreover, p∗ ∈ U is stable equilibrium of X and X(x) ≠ 0 if x ≠ p∗ . Construct an auxiliary vector field X̃ : ℝ2 → ℝ2 with the properties: X̃ is Lipschitzcontinuous with Lipschitz constant L, X = X̃ in some neighborhood of p∗ . Yorke’s ̃ result can be applied to deduce that every non-constant periodic solution of ẋ = X(x) 2π 2π ̃ p), where x(t, ̃ p) is the has period T ≥ L . Let us fix τ ∈ ]0, L [ and define h(p) = x(τ,

172 | 7 Solutions to the exercises ̃ ̃ p) = p. Then h satisfies the conditions (h1) and (h2). solution of ẋ = X(x) with x(0, The equilibrium p∗ becomes a stable fixed point of h. The key observation is that p∗ is isolated in Fix(h) thanks to Yorke’s result. Theorem 12 can be applied to deduce that I(h, p∗ ) = 1. Now Lemma 14 implies that the index of X̃ at p∗ is (−1)d . From the excision property we deduce that this is also the index of the original vector field X at p∗ . Exercise 54 It is not restrictive to work with the open unit disk D = {p ∈ ℝ2 : ‖p‖ < 1}. Select R < 1 x such that cl(Ω) ⊂ {p ∈ ℝ2 : ‖p‖ < R}. Consider the map M : ℝ2 → D, M(x) = m(‖x‖) ‖x‖ where m : ℝ → [0, 1[ is continuous, increasing, m(r) = r if r ≤ R and limr→+∞ m(r) = 1. ̂ Then h = h ∘ M belongs to ℰ (ℝ2 ) and h(Ω)̂ = h (Ω)̂ = h? (Ω) = h(Ω). 1

Exercise 55 ẋ1 = x2 , ẋ2 = −ω2 x1 , ω =

1

2π , 7T

1

φ ≡ (0, 0).

Exercise 56 See the second example in Section 2.4 with φ ≡ E. Exercise 57 Let M > 0 be a constant such that ‖X(x)‖ ≤ M for each x ∈ ℝ2 . Fix τ > 0 and define h = ϕτ . We know that h ∈ ℋ∗ (ℝ2 ). Next we prove that Fix(h) = {0}. In principle h could have fixed points corresponding to periodic solutions of period τ. This is not the case because the system has no closed orbits. Assume by contradiction that Γ ⊂ ℝ2 is a closed orbit, then Ri (Γ) should contain an equilibrium. The origin is the only equilibrium and therefore 0 ∈ Ri (Γ). The orbit associated to the solution given by (iii) is denoted by R; it should connect the two components of ℝ2 \ Γ. In consequence R and Γ should cross and this is impossible because they are two different orbits of the same system. Once we know that the origin is the only fixed point of h, we can easily verify that the three conditions in Theorem 13 hold. The only delicate point is in condition (ii). For every solution of the system, ‖x(t) − x(s)‖ ≤ M|t − s|,

t, s ∈ ℝ.

(7.1)

Given a solution x(t), there exist a number b > 0 and a sequence tk → +∞ such that ‖x(tk )‖ ≤ b. Select integers nk → +∞ with nk τ ≤ tk < (nk + 1)τ. Then, using (7.1), ‖x(nk τ)‖ ≤ b + Mτ. Theorem 13 is applicable and p∗ = 0 is g. a. s. with respect to the map h. Using again (7.1) we conclude that the origin is also g. a. s. for the system of differential equations. The alternative proof using Poincaré–Bendixson theory goes as follows. The flow {ϕt } is globally defined on 𝕊2 = ℝ2 ∪ {∞} with ϕt (∞) = ∞. In parallel to Exercise 47

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| 173

one can prove that x = 0 is g. a. s. if and only if 0 ∈ Lω (p) for each p ∈ ℝ2 . This limit set is understood on the Riemann sphere. By a contradiction argument assume that 0 ∈ ̸ Lω (p) for some p. Then either Lω (p) = {∞} or Lω (p) contains a homoclinic orbit tending to ∞. Both possibilities are incompatible with (ii). Exercise 58 In cylindrical coordinates define θ̇ = 1,

ṙ = −

(r 2 − 1)2 r, (r 2 + 1)3

ż = −

z . 1 + z2

The origin is not g. a. s. because the region of attraction is r < 1. A solution (in Cartesian coordinates) satisfying the condition (iii) is x(t) = (0, 0, x3 (t)) with x3 (0) > 0. Exercise 59 bt u(t) ≤ u0 ebt , v(t) ≤ v0 eϕ(u0 e ) if t ∈ [0, ω[. Exercise 60 From the additivity of λ, λ(G) = λ(G \ h(G)) + λ(h(G)). Since λ(h(G)) = λ(G), we deduce that G \ h(G) has zero measure. In particular this set has empty interior. Since h is open, the set h(G) is open and G \ h(G) is closed relative to G; that is, G \ h(G) = C ∩ G with C closed. We express G as a countable union of closed sets G = ⋃n≥0 Fn . Then G \ h(G) = ⋃n (C ∩ Fn ), where each C ∩ Fn is closed and has empty interior. Exercise 61 The only difference with respect to the proof of Lemma 17 is in the way to show that the inclusion G ⊂ cl(h(G)) holds. Define W = G \ cl(h(G)). Then W ∩ h(G) = 0 and h(W) ⊂ h(G). More generally, hn (W) ⊂ hn (G) ⊂ h(G) for each n ≥ 1. In consequence, the open set W satisfies W ∩ hn (W) = 0 for each n ≥ 1. We conclude that W is empty. Exercise 62 − 1 1 f (x, y) = (x + e x2 sin( x1 ), 21 y), Fix(f ) = {( nπ , 0) : n ∈ ℤ \ {(0, 0)}} ∪ {0}. Exercise 63 ̈ Let x(t) be a 2π-periodic solution. From the equation, |x(t)| ≤ |λ| + 1 for every t. Inte2π x(t) dt = 0, implying x(t0 ) = 0 for some t0 . grating over a period in the equation, ∫0 1+x(t) 2 t

̈ ̇ 1 ) = 0. Then x(t) ̇ = ∫t x(s)ds Since x(t) is 2π-periodic, there exists some t1 such that x(t t

̇ ̇ implies |x(t)| ≤ π 2 (|λ| + 1). implies that |x(t)| ≤ π(|λ| + 1) and x(t) = ∫t x(s)ds 0

1

174 | 7 Solutions to the exercises Exercise 64 Assume that (q(t), p(t)) is a periodic solution. From the former equation, q̇ 1 =

𝜕H 2 = q12 + p21 + (3p21 + 1)(q22 + p22 − 1) 𝜕p1

we deduce that q̇ 1 (t) ≥ 0. Since the function q1 (t) is periodic and has non-negative derivative, it must be constant. In particular, q̇ 1 = 0 and q1 = p1 = 0, q22 + p22 = 1. It is easy to prove that these equations define a closed orbit associated to the solution q1 = p1 = 0, q2 + ip2 = eit .

7.5 Chapter 5 Exercise 65 Assume by contradiction that h(𝒩 ) ∩ 𝒩 ≠ 0 and h, g are strongly equivalent. Let D be a modification disk, since h and g coincide outside D, the set 𝒩 must be contained in D. Hence, h(𝒩 ) ∩ 𝒩 ≠ 0 implies h(D) ∩ D ≠ 0 and this is against the definition of D. Exercise 66 Since β󸀠 = [0, 21 ] × {0} is contained in the interior of the disk ψ(D), we can find ϵ ∈ ]0, 1[ 󸀠 such that the neighborhood β2ϵ is also contained in int(ψ(D)). Here 󸀠 = {p ∈ ℝ2 : dist(p, β󸀠 ) ≤ 2ϵ}. β2ϵ

Let f : ℝ2 → ℝ be the function defined by f = id outside [−ϵ, 21 + ϵ] × [−ϵ, ϵ]. For each |y| < ϵ the function f (⋅, y) is the piecewise linear function satisfying f (−ϵ, y) = −ϵ, , f ( 21 + ϵ, y) = 21 + ϵ. Define ϕ0 : x1 = f (x, y), y1 = y. It is easy to f (0, y) = 21 − |y| 2ϵ check that ϕ0 belongs to ℋ(ℝ2 ) and ϕ0 = id outside ψ(D). Moreover, ϕ0 (0, 0) = ( 21 , 0), ϕ0 (1, 0) = (1, 0) and ϕ0 ([0, 1] × {0}) = [ 21 , 1] × {0}. Exercise 67 In the first case ν = +∞. In the second case hk (α) = {eiθ : kΘ ≤ θ ≤ (k + 1)Θ} and }. ν = min{n ∈ ℤ : n ≥ 2π Θ Exercise 68 Assume by contradiction that q ∈ hν−1 (α). Then p = h−1 (q) ∈ hν−2 (α). In consequence α ∩ hν−2 (α) ≠ 0. This is incompatible with the definition of the index ν if ν > 3. Let us now consider the case ν = 3. From p ∈ h(α) we deduce that p ∈ (α \ {q}) ∩ h(α). Since α is a translation arc, p = h(q) and we find again a contradiction with the definition of ν.

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| 175

Exercise 69 In the case p∗ = p it is sufficient to apply Proposition 13. In the case p∗ = hν (p) only Proposition 12 is applied. In the case p∗ = p = hν (p) we can take h = g, Γ = α ∪ h(α) ∪ ⋅ ⋅ ⋅ ∪ hν−1 (α). Exercise 70 This is straightforward. Exercise 71 1 h(tp) if t‖p‖ ≥ 1, Ht (p) = { t S(p) if t‖p‖ ≤ 1.

Exercise 72 We replace β by β̂ and work with the corresponding space X.̂ Consider the map f : 𝔼 → ℝ2 ,

f (x1 , x2 ) = (x1 , x2 −

x2 √x12

+ x22

).

It is not hard to check that each ray {λ(x1 , x2 ) : λ ≥ 1} emanating from a point (x1 , x2 ) ∈ 𝕊1 is transformed into a parallel ray emanating from (x1 , 0). The sense of this ray depends upon the sign of x2 . From this we deduce that the image f (𝔼) is precisely ℝ2 \ β.̂ Define the bijection between 𝔼 and X,̂ ψ : 𝔼 → X,̂

f (x) if x ∈ (𝔼 \ 𝜕𝔼) ∪ {(1, 0), (−1, 0)} ψ(x) = { (f (x), sign(x2 )) if x ∈ 𝜕𝔼, x ≠ (±1, 0).

Then d(ξ , η) = ‖ψ−1 (ξ ) − ψ−1 (η)‖ if ξ , η ∈ X.̂

7.6 Chapter 6 Exercise 73 Ω1 = {x ∈ ℝd : ‖x‖ < 1}, Ω2 = {x ∈ ℝd : 0 < ‖x‖ < 1}, ℳ⋆ (Ω2 ) = {f ∈ ℳ⋆ (Ω1 ) : f (0) ≠ 0}. Exercise 74 Define H(x, t) = (1−t)f (x)+tg(x). If f = g on 𝜕Ω, H(x, t) = f (x) ≠ 0 for each t ∈ [0, 1] and x ∈ 𝜕Ω. Assume now that ‖f (x) − g(x)‖ < ϵ if x ∈ 𝜕Ω, where ϵ := minx∈𝜕Ω ‖f (x)‖. Then, for each t ∈ [0, 1] and x ∈ 𝜕Ω, ‖h(x, t)‖ ≥ ‖f (x)‖ − t‖f (x) − g(x)‖ ≥ ϵ − ‖f (x) − g(x)‖ > 0.

176 | 7 Solutions to the exercises Exercise 75 When a < 0 < b the first case is a consequence of the properties of normalization and homotopy. When 0 < a < b we extend f to f ⋆ : [−a.b] → ℝ, f ⋆ = f (a) on [−a, a]. Then deg(f ⋆ , ]−a, b[) = 1 and the additivity property implies that deg(f ⋆ , ]−a, b[) = deg(f , ]a, b[). The first case is proved. In the third case f (a) and f (b) have the same sign and the function f is homotopic to the constant function g ≡ f (a). Since g has no zeros we conclude that deg(f , Ω) = deg(g, Ω) = 0. In the latter case f (a) > 0 > f (b) we extend f by symmetry f ⋆ : [a, 2b − a] → ℝ, f ⋆ (x) = f (2b − x) if x ∈ [b, 2b − a]. Since f ⋆ (a) = f ⋆ (2b − a) ≠ 0 we are in the third case and deg(f ⋆ , ]a, 2b − a[) = 0. By additivity, deg(f ⋆ , ]a, 2b − a[) = deg(f , ]a, b[) + deg(f ⋆ , ]b, 2b − a[). Since f ⋆ (b) < 0 < f ⋆ (2b − a) the last degree is in the first case and deg(f ⋆ , ]b, 2b − a[) = 1. Exercise 76 Define ϕ : [−1, 3] → ℝ, −ξ if ξ ∈ [−1, 1], ϕ(ξ ) = { ξ − 2 if ξ ∈ [1, 3], and f (x1 , . . . , xd−1 , xd ) = (x1 , . . . , xd−1 , ϕ(xd )). On the domain Ω̃ = {x ∈ ℝd : |xi | < 1, i = 1, . . . , d − 1, xd ∈ ]−1, 3[} the map f is linearly homotopic to g(x1 , . . . , xd ) = (x1 , . . . , xd−1 , 1), a map without zeros. Then deg(f , Ω)̃ = deg(g, Ω)̃ = 0. By additivity deg(D, Ω) = − deg(f , Ω1 ) where Ω1 = {x ∈ ℝd : |xi | < 1, i = 1, . . . , d − 1, xd ∈ ]1, 3[}. By excision, deg(f , Ω1 ) = deg(f ⋆ , Ω)̃ where f ⋆ (x1 , . . . , xd ) = (x1 , . . . , xd−1 , xd − 2). Then H(x, t) = (x1 , . . . , xd−1 , xd − 2t), t ∈ [0, 1], is an admissible homotopy between f ⋆ and id on Ω.̃ Exercise 77 From previous discussions we know that if 0 ∈ Ω then deg(L, Ω) = deg(id, Ω) if det L > 0 and deg(L, Ω) = deg(D, Ω) if det L < 0. Here D is the linear map in Exercise 76. Exercise 78 By excision we can assume that Ω is a ball centered at x∗ . Define 1 f (tx + (1 − t)x⋆ ) if t ≠ 0, H(x, t) = { t 󸀠 f (x⋆ )(x − x⋆ ) if t = 0.

This is a homotopy between f and the affine map A(x) = f 󸀠 (x⋆ )(x − x⋆ ). Exercise 79 The formula is clearly valid for f of class C 1 with non-degenerate zeros. In the general case we pass to the limit.

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| 177

Exercise 80 Approximate f1 and f2 by C 1 functions with non-degenerate zeros and observe that det[(f1 × f2 )󸀠 (x, y)] = det[f1󸀠 (x)] det[f2󸀠 (y)]. Exercise 81 Let us fix a positive number δ > 0 such that the ball B(x0 , δ) is contained in Ω. We start with a preliminary property: Given ŷ ∈ Ω and t ∈ ]0, 1[, define xt = (1 − t)x0 + t y.̂ Then B(xt , (1 − t)δ) ⊂ Ω. See Figure 7.3. To prove this claim we take an arbitrary point ξ ∈ B(xt , (1 − t)δ). The inequality ‖xt − ξ ‖ < (1 − t)δ is equivalent to ‖z − x0 ‖ < δ with 1 t z = 1−t y.̂ Then z ∈ B(x0 , δ) ⊂ Ω. The two points z and ŷ belong to Ω and, by ξ − 1−t convexity, also ξ = (1 − t)z + t ŷ is in Ω.

Figure 7.3: A decreasing ball inside Ω.

Once we have proved this preliminary property we are ready to complete the exercise. Let us take y ∈ cl(Ω) and t ∈ [0, 1[. We select a sequence ŷn ∈ Ω converging to y and define xn = (1−t)x0 +t ŷn . The ball B(xn , (1−t)δ) is contained in Ω and it must be noticed that the radius of this ball is independent of n. We claim that, for large n, the point (1−t)x0 +ty belongs to this ball and hence to Ω. Indeed, ‖(1−t)x0 +ty−xn ‖ = t‖y− ŷn ‖ → 0 as n → +∞. We can assume t‖y − ŷn ‖ < (1 − t)δ for large n. Exercise 82 If F has fixed points on the boundary the conclusion is obvious. Otherwise the conditions of Brouwer’s principle hold.

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Index Alexander’s isotopy 139 analytic sets 117 area-preserving 59, 113, 118 asymptotic fixed point theorem 155 asymptotically periodic solution 6, 17 asymptotically stable equilibrium 78 asymptotically stable fixed point 94 autonomous system 78

Lotka–Volterra system 4 Lyapunov stability 73, 124

bounded in the future (solution) 7 Brouwer’s lemma 47, 56, 62, 145

pendulum equation 30 periodic solution 6, 26, 75 perpetual stability 74, 115 persistence 30, 104 plug 83 Poincaré map 4, 8, 41, 74 Poincaré–Bendixson theorem 20 Poincaré’s recurrence theorem 62 population dynamics 110 positive periodic solution 29 positively invariant 113 prey–predator system 29, 122

closed orbit 81, 124 covering map 64 diffeotopy 10 difference equation 5, 19 equilibrium 78 extinction 18, 111 first Massera theorem 18 fixed point 46, 59, 63, 75, 120 fixed point index 93, 147, 154 fixed point set 42, 52, 138 forced pendulum equation 1 free embedding 43, 56 freely equivalent 131 globally asymptotically stable 106 Hamiltonian system 24, 118 index of a vector field 78 instability criteria 101 invariant curve 143, 155 invariant neighborhood 92, 97 involution 52 isochronous 32 isotopy 9, 140 Jordan–Schönflies theorem 144 limit set 6, 42, 81 locally free map 44

Massera’s theorem 68 orbital stability 124 orientation-preserving embeddings 40 oscillator 1, 122

recurrent point 46 restricted three body problem 2 second Massera theorem 25 Sitnikov problem 2 stability of fixed points 74 stable fixed point 93, 113 standard map 13 strongly equivalent 131 sub-harmonic solution 6 suspension bridge 2 topological degree 147 topological disk 39 topological embedding 37 translation arc 46, 48, 132, 134, 145 trivial dynamics 42 unbounded solutions 21 Van der Pol equation 1 weak involution 53

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